I really thought it was impossible till I did the math. Obviously vertical is 9+9. Then I said the small horizontal is x and the bottom is 12-x. Put it all together and the x's drop out. And the solution is the answer to life, the universe, and everything!
At first read I thought they were asking for the area 😂, I read the question and though, oh, that's easy: consider the inner horizontal segment be 0 and that's it.
Nerdy point here: 42 is not the answer to Life, the Universe, and Everything (nor is it the answer to the Meaning Of Life which some people posit). It is instead the answer to The Ultimate Question Of Life The Universe And Everything, which has not yet been asked. We don't know what the Ultimate Question even is.
@@zuggo The assumption was that as Arthur was a human from Earth, he must have The Question buried in his brain laying dormant, and they just needed to tease it out. However the result was not helpful, and it was decided that was probably not The Ultimate Question after all.
It's actually quite simple if you look at it algebraically. On the left side, we already have 9, and the right side has no overlaps, and all angles are 90 degrees, meaning that the three segments must add up to 9 to match the left side. So you don't even need to know the values of each individual line. Then the top we do have an overlap, so we can't do the same thing. However, we can give the short segment a value of x. So, the bottom line will be 5+7-x, because x is the overlap. And for the top, because we're checking the perimeter, we have to include all three, so it's 5+7+x. So, all in all it's 9+9+5+7+x+5+7-x. Add up all the numbers and you have 42+x-x. Well, now the x's cancel each other out, leaving only 42 for the answer, so you don't even need to know the value of x.
Call the short horizontal side x. It follows that all horizontal lines contribute 5 + x +7 + (5-x) + 7 = 5 + 7 + 5 +7. All vertical lines contribute 9 + 9. The total is 42
They aren't drawn to scale. The only thing why it works is because the lines are straight. Solving this on your own gives you alot better of an understanding of the problem
Not that i care much about the colorblind, but it also looks more childish and less math-y. We're making formulas ans algebra, just use math instead of colors like its for babies
There are four horizontal edges: one length 5, one length 7, the one between them (call it A) and the one at the bottom (call it B). If you increase length of B, you decrease the length of A by the same amount, so the perimeter stays the same. If you continue this until A has length zero, the bottom edge has length 12.
I had to watch that twice to get his point about the horizontal lengths. I think this way would be easier: Call the length he marked in purple, x. The total horizontal lengths are the top line (5), plus the purple line (x) plus the orange line (7), plus the bottom line (5 + 7 - x). 5 + x + 7 + (5 + 7 - x) = 24. (The x's cancel). 24 + the vertical lines (18) = 42.
I feel like the yellow-purple-orange line could use a bit more explanation, I got it after staring for a few mins but as presented it feels like a jump in logic.
We know that yellow + purple = 5 and orange = 7. The bottom line is equal to yellow + orange. Then we need to add the small purple part that “goes” inwards. We add 5 + 7 a second time and without knowing the exact numbers we are able to determine the perimeter of the shape.
so lets say the purple line is length x. The bottom line will be 5 + 7 - x, cause that length x is included in both the 5 and the 7, so it's counted twice and we only need it one time. So now, if you go around and add all the lengths, you get: 9 + 5 + 9 + x + 7 + (5 + 7 - x), where the second 9 is the sum of the right hand vertical lengths. You see that in that result, both x values are canceling each other, and you get the remaining 9 + 5 + 9 +7 + 5 + 7 = 42
It might have been better to use a different color than yellow, since that had already been used for the vertical lines. By making it a different color, it becomes clearer that the perimeter consists of two sections each of six different colors, and therefore the sum is 2x(9+5+7).
It could help to use some Algebra. x is the part of the side of length 5 (or "line A" to simplify the rest of this explanation) that's _not_ included in the side of length 7 (or "line B"). y is the part of A that _is_ included in B. z is the part of B that's not included in A. The bottom line (line C) is the sum of x, y, and z. There's also an additional y from where the shape moves left before moving down and right again. x + y = 5 (line A) y + z = 7 (line B) x + y + z = ? (line C) Perimeter: 9 + 9 + 5 + y + 7 + (x + y + z) (Parentheses around the line C part). Substitute the first y with (5 - x). Substitute the second y with (7 - z). 9 + 9 + 5 + (5 - x) + 7 + x + (7 - z) + z x and z cancel. 9 + 9 + 5 + 5 + 7 + 7 = 42
Just visually stretch out the bottom horizontal line until the indentation goes away: 5+7=5+7. The right-side verticals are pretty obvious: whatever they are = 9. 12+12+9+9=42
Yes, that's essentiallyhow I did it too. My visualization was to slide the top left part (sides labeled 9 and 5) leftward while keeping the unlabeled bottom and right sides in place. While doing this, all but two sides keep their lengths. The purple side gets smaller, while the bottom side gets larger by the same amount. Continue until the purple side has reached length 0, so that the overall shape is that of an L. Or a rectangle with the top right corner bitten off. Biting off a rectangular corner from a rectangle does not change the perimeter, which is quite obvious if you have drawn the image. So the solution is simply the perimeter of the full rectangle, which has horizontal sides of length 5+7 and vertical sides of length 9. So the overall perimeter is 2 x (5+7 + 9) = 42.
You're showing that the limit as x->0 of the perimeter of this shape is 42. What you haven't shown is that the perimeter is indifferent to the value of x.
@@rickdesperNo. You're taking the purple line x out and putting it in the bottom edge. This doesn't alter the perimeter and only simplifies the shape.
The equals sign and normal rectangle threw me off so much because it made me think I was somehow supposed to be manipulating the shape into a rectangle first and then finding the perimeter, as soon as I got that idea out of my head it was much easier.
4:12 "Life, the universe, and everything" isn't a question, so can't even have an answer. 42 is the answer to the ultimate question about life, the universe, and everything.
@@fopdoodler9427 How do you figure that? Not only have I read it, multiple times, in multiple languages, but apparently you haven't read it *and* are too dim to even understand basic English. Unlike Mr Adams, who's a master of it. Your comment is almost, but not quite, completely unlike a sane one, in much the same way that mine wasn't.
@Grizzly01-vr4pn I did read it, my dear sir. And I just checked online copies to see where it's written and actually found it again, that's why I gave a page number. It won't be the same page number for every copy, but page numbers are an indicator.
Although it's nice to get to an answer by only ADDING all the bits, I think a more satisfying way is to say that the UPPER horizontal lines add to 5+7+x (where x is the purple bit), and the base line comes to 5+7-x (ie, this time x SUBTRACTS). Then, when we add all the bits together, the x will cancel out (x-x), and so we get to a numeric result. I think this response helps us to better understand intuitively WHY we can get to a result even when we don't know how long x is.
Might have been more interesting at the end to slide the 7 length horizontally (along with the adjoining sides), and show that the perimeter remains 42 with that translation as well.
I think it means they are both rectilinear by definition, i.e contained by, consisting of, or moving in a straight line or lines, despite the left polygon being a composite of different shapes.
I guess this was the teacher's reminder of the solution path the students were presented in class. It may have made sense to the students, but without such context it's too obscure and in fact seriously misleading. Here is my solution path, which retroactively made sense of the hint: Slide the top left part of the shape (sides labeled 9 and 5) leftward while keeping the unlabeled bottom and right sides in place. While doing this, all but two sides keep their lengths. During this process, the side painted purple in the video gets smaller, while the bottom side gets larger by the same amount. Continue until the purple side has reached length 0, so that the overall shape is that of an L. Or a rectangle with the top right corner bitten off. Biting off a rectangular corner from a rectangle does not change the perimeter, which is quite obvious if you have drawn the image. So the solution is the perimeter of the resulting full rectangle, which has horizontal sides of length 5+7 and vertical sides of length 9. So the overall perimeter is 2 x (5+7 + 9) = 42. This resulting full rectangle is probably what the rectangle on the right hand side of the equation was supposed to be.
@@IKVKronosOne In this video "rectilinear" is being used to refer to a shape where all angles are right (a mix of 90° and 270° angles, as applicable), since polygons are already defined to be made of straight line( segment)s. Or equivalently, a shape where any two sides are either perpendicular or parallel to one another.
I'm used to so much "Not drawn to scale" that honestly i just assumed it wasn't drawn to scale. honestly i think problems like these should always require mentioning whether or not they're drawn to scale.
Don't you assume they're not drawn to scale unless specified otherwise? . . . are they ever specified as drawn to scale unless you're exploring something empirical and using rulers or whatever to demonstrate the rule?
I don't think it necessarily is drawn to scale. I'm pretty sure you can assign any values you want that are less than 5 to the horizontal unknown and assign any values to the three vertical unknowns, as long as they add up to 9, and it will always be 42.
Regarding the last part of video where you extend the vertical green line: That's how I solved it. 1. Extend the green line up and down until it achieves a length of 9, forming a standard rectangle with dimensions 9 by x. 2. Double the two edges that stick out as these are "rectangles" with width 0. So this part of the perimeter is 14 and 2(5 - x) 3. The perimeter of the large rectangle is 18 + 2x 4. Total P = 14 + 2(5 - x) + 18 + 2x = 14 + 10 - 2x + 18 + 2x = 14 + 10 + 18 = 42
I got confused at around 3:15 when I didn't realise the 2 * "Yellow + Orange = 5+7" contains the two extra "Purples" in the middle, maybe this could have been made more obvious at this point by pointing it out.
Yeah, I got that eventually, it just took a while for me to click that the two purple segments in the middle are covered by that. I.E. that the top part is "Yellow + Orange" (and not Yellow + Orange + Purple), the bottom part is also "Yellow + Orange". My argument is that it would have been better if Presh specifically pointed this out when he was rearranging the colored segments (if he did, I completely missed that).
@@caoryn It's because he very clearly says that we know that yellow+purple+orange "must equal 5+7," which, of course, they don't. It's just that the unknown length of purple cancels out when you add up all the sides.
0:12 you haven't even asked the question yet and I'm sitting here with the answer to life, the universe, and everything. Will you mention that? Edit: YOU WILL!
I work in a machine shop and have to do a lot of this with different parts because on the technical drawing not all dimensions are directly given some are often implied.
The only thing I would say is I had the a+b+c = 9 But I saw and solved the horizontal portion as 5 + 7 which were given but I broke it down in (7-5) and (5-(7-5)) for the portions where there is overlap on the 5 and 7 horizontal lines so that it was 9+5+a+(7-5)+b+7+c +7+(5-(7-5)) Where a+b+c =9 or 9+9+7+7+5+(5-(7-5))+(7-5) which once I saw the video I realized how it really could be written more similar how he did as 9+9+7+7+5+5 because all I did was break the second 5 down to solve for each of those individual lengths
my silly ass thought i was supposed to find the numbers for all of the sides one at a time but you oculd of just, skipped them entirely? this video enlightened me lmao
Lets say we moved the green line to the right lengthening the bottom line. The bottom line lengthens the same amount as the line at the top of the green shorten so we could make it so that the 7 starts where the 5 ends. This essentially would make it about calculating the perimeter of a rectangle.
To be more exact, one would need to move not only the green line to the right, but the other vertical lines as well in order to end up with a rectangle. First move both the green and the bottommost vertical line to the right by the same amount until the green line joins the topmost vertical line to become a single edge of the shape. Then move that edge to the right until the shape becomes a rectangle. There are multiple simple ways to solve this problem and any solution will seem easy to some people but difficult to others.
Xs dont cancel. You make no sense. The guy in the video trolled all of you. You cant solve for X. You are ignoring the middle line of X. Top Line = (5+7-X) Middle Line = X Bottom Line = (5+7-X) The real problem is 9+9+(5+7-X)+X+(5+7-X) = y [I added in parenthesis to make it easy to read] 42-X+X-X = Y This is your final answer 42-X=Y For this guy to get y = 42 then x = 0 which cant be true by looking at the picture. You have to know the value of X as X does not equal 0. The entire perimeter cannot be solved util you find the value of X which you cant do. You can only get 42-X=Y
@@common_c3nts What are you talking about? There are not three horizontal edges, but four. From top to bottom, they have length 5, X, 7, and 5+7-X. The sum of these lengths is 24. No dependency on X left. The lengths of the vertical edges sum to 18, giving a total perimeter of 42.
A very intuitive way to present it. I ended up doing a lot of extra work making equations for everything I didn't know, adding it all up, then substituting until I got the answer.
Xs dont cancel. The guy in the video trolled all of you. You cant solve for X. You are ignoring the middle line of X. Top Line = (5+7-X) Middle Line = X Bottom Line = (5+7-X) The real problem is 9+9+(5+7-X)+X+(5+7-X) = y [I added in parenthesis to make it easy to read] 42-X+X-X = Y This is your final answer 42-X=Y For this guy to get y = 42 then x = 0 which cant be true by looking at the picture. You have to know the value of X as X does not equal 0. The entire perimeter cannot be solved util you find the value of X which you cant do. You can only get 42-X=Y
His explanation lacked a little clarity. The trick is in that the horizontal distances are given from opposite ends. I had to do it algebraically myself to see how the center horizontal distance cancelled.
Perform a geometric transformation that does not affect the perimeter: shift the vert. green line, the hor. white 7-line and the vert. yellow line to the left until the green line meets the vert. white 9-line. The hor. purple line will grow and the hor. bottom line will shrink, and both changes will compensate each other. Then the perimeter of the shape is clearly: 2*5+2*7+2*9=2*21=42
I knew intuitively that it was 42-but your color coding was a great way to make it clearer for people who aren’t as nimble in visualizing such things on their own (such as the supposed “baffled parents”). Well done!!
I let the purple length equal x. The horizontal segments are now 5, x, 7, and the bottom is 5+7-x, as we have to remove the overlap. The sum of the horizontal segments is 5+x+7+5+7-x. That sums to 24 as the x values cancel each other. Now add 18 for the vertical segments to get to the solution of 42.
I was so convinced that this was so much more complicated than it was that when I did the step as you described I couldn't believe it was that simple. The curse of knowledge I suppose.
I’m surprised in the final part of the video, when you changed the shape by moving the rectangles up and down (but maintained the same perimeter) you didn’t also demonstrate that you can also adjust the horizontal width of the shape (maintaining 5 and 7, but changing the amount those lengths overlap), and still have the same perimeter…
You can even make a rectangle out of it by setting green, blue and violet to zero, hence getting a rectangle of sizes 9 & 12, then getting 42 is trivial
Exactly, that is how I solved it in my head. There is the freedom to shift the horizontal 7 to the right, until it is easy to see the horizontal lines together must have length 2 x (5+7).
The horizontal perimeter is given by the sum of bottom + the sum of the 3 other horizontal segments. The bottom = 5 + 7 - purple; the 3 segments = 5 + 7 + purple. Together you get the horizontal perimeter to be 5+7-purple+5+7+purple. The purples cancel and you get 5+7+5+7=24. Add the 9x2 vertical and you get 42.
@@raifmt Right. If you think about making a rectangle, then of course the top and bottom must be equal. I guess it was easier for me to think about adding up all 4 horizontal segments. But your approach absolutely works too.
I could quickly see that all the unknown vertical sides must add up to 9. It took me a long time to understand how to deal with unknown horizontals. Having read a lot of other people's solutions I have come up with my simplest way of dealing with horizontals: - notice that shrinking the shorter unknown horizontal, and stretching the longer one by the same amount does not change the perimeter, - shrink the shorter to zero (and stretch the longer) - result is an L shape, where it easy to see that the bottom side is 7+5 = 12 Verticals add to 2 x 9 =18 Horizontals add to 2 x 12 = 24 18 + 24 = 42 One slight mental block I had was trying to keep the area the same, whereas that was not relevant. Also the feeling that I could do this easily if I was 50 years younger.
Its dishonest if the question is written to assume the lines are drawn to scale and the user can use a measuring stick without explicitly indicating such in the problem statement. A lot of times, problems like these are purposely not drawn proportional to the given values.
@@Jeremo-FD Ya, you're right. I had initially thought that he was implying the horizontal yellow and purple were equal parts so 2.5 + 2.5 + 7 on the horizontals. When it actually looks like he was implying some theorem about rectumlinear proportionality with the sum of the lengths for 5 and 7. It makes more sense to think of the sum of the yellow and purple as 5 with the two unknowns not being important. Then of course the orange being 7 because of the rectumlinearness.
@@Commenter-f8bPresh's explanation certainly was lacking. He didn't specify *how* he chose the yellow and the purple part. So I see how you got confused here.
This problem is kind of nice, but as usual, there's no single explanation that is absolutely clear to everyone. This particular problem would have been a good opportunity to show different approaches. Some I could think of at the top of my head: 1) Compare edge lengths visually like in the video. 2) Name the edges and do it algebraically. 3) Change the shape in a "continuous" fashion that does not change the perimeter until you end up with a rectangle. E.g. take the 7-edge and its two adjacent vertical edges and move them to the right until the "purple" horizontal edge disappears, leaving with an L shape. Then move the 7-edge up until you have a rectangle. 4) Change the shape by repeatedly placing parts of it at different places until you have a rectangle. E.g. rotate the three "indent" edges by 180° so that they now form a "bulge" instead of an indent. Then get rid of the 270° angles by taking the the two incident edges and again rotate them by 180° so that they no longer point "inwards". 5) Follow the outline of the shape until you reach you starting point again and observe that for both the vertical and the horzizontal edges, the ones whose lengths we know are traversed in one direction, while all the others are traversed in the opposite direction.
(3) Making the “L” is easiest. The length of the purple inside horizontal segment is arbitrary, so just choose zero and make the “L”. The perimeter of an “L” is same as the perimeter of the rectangle of same base and height, because the two “inside segments” are identical to the original upper right of the rectangle. Just kind of dented in. Anybody with a modicum of geometric visualization ability will see it pretty fast. And it’s a very satisfying, elegant answer.
Really cool challenge. Thanks a lot! A little bit of pendantry though: 0:56 a rectangle with a semicircle at two ends is not called a pill. In geometry pill is a three dimensional shape. Cut that shape in half and rotate it along a circle and you have a pill. (The shape is commonly used as a collision volume in virtual reality and for physics simulation btw. It's one of the four volumes that is easiest for a physics engine to handle - the other three are the cube, the sphere and the cylinder.) The two dimensional shape shown in the video is usually called a stadium. I've never heard the name trackhouse bedore but it does fit.
Wikipedia would disagree with you. > A stadium is a two-dimensional geometric shape constructed of a rectangle with semicircles at a pair of opposite sides.[1] The same shape is known also as a pill shape,[2] discorectangle,[3] obround,[4][5] or sausage body.[6] While the 3d shape is called a capsule.
@@woobilicious. Oh, maybe I was wrong then. I only know this from working with 3D graphics. But in that case, we should probably avoid using the "pill" name at all if it's ambigious. The most common name for the 2D variant is still "stadium" and the 3D one is also commonly known as a "capsule". Those two words are unambigious in geometry so prehaps we should stick to them.
Not so. It helped me solve it visually. The composite rectilinear shape can be converted to the rectangle. You move the 7 horizontal line to the top to meet the 5. All of the vertical lines are moved to the right side edge. Then the leftover short horizontal line must fill in the bottom edge to complete the rectangle. Now you have a 9x12 rectangle.
Got it right! I did mine a bit different as I broke the 7 up as the overhang part must be the difference between 7 and 5 so I ended up with the top horizontal line being (5+5+2+2) leaving the bottom line 5+5. We arrived at the vertical lines values of 9 each the same so (9x2)+(5x2)+(5+5+2+2)=18+10+14= 42 The distributive property of addition is great!
Consider the horizontal lines. Relocate the 7 to join onto the 5. This leaves the unknown short length. By inspection the right hand end of the 7 has moved right by the length of the short unknown so this unknown can fit onto the right hand end of the base, making the base 7 + 5. = 12. So we have two equal horizontal lengths of 12 to add to two vertical lengths yielding the equivalent perimeter.
1:51 Shift the green line right, to be under the blue, sliding the 7 line and yellow line the same distance. The bottom line is lengthened the same length as the middle horizontal that is eliminated. Easily visualized as 5+7+12 + 9+9
Professor of computer science here. While I appreciate showing how to solve this graphically. I think you could have been clearer that the reason this can be solved is because there is exactly one unknown variable. The overlap between the side of length 5 and the one of length 7 and that the perimeter can be calculated by adding that value and then subtracting it again. Thus eliminating the variable. A shape with extra variables would not be solvable. Writing this out algebraically might also have been a good idea and I suspect that would give students an understanding about how to solve this problem more generally.
Xs dont cancel. You make no sense. The guy in the video trolled all of you. You cant solve for X. You are ignoring the middle line of X. Top Line = (5+7-X) Middle Line = X Bottom Line = (5+7-X) The real problem is 9+9+(5+7-X)+X+(5+7-X) = y [I added in parenthesis to make it easy to read] 42-X+X-X = Y This is your final answer 42-X=Y For this guy to get y = 42 then x = 0 which cant be true by looking at the picture. You have to know the value of X as X does not equal 0. The entire perimeter cannot be solved util you find the value of X which you cant do. You can only get 42-X=Y
I disagree, the verticals are not a problem as those are all right angles so the verticals on the right must add up to the vertical on the left. That's fine. However, we started out with 5 unknown sides based on the problem, 3 of which are verticals that must add up to the right hand side of 9, giving us 2 unknowns without the information to solve it. Even if we divide the bottom the way the video does, that still gives us a section that cannot be figured out with the information provided. You'll note that he didn't even try to lengthen or shrink that bottom segment. The video errs in assuming that the bottom is 9 due to scale. That cannot be assumed unless it is stated to be the case.
@@common_c3nts incorrect. You do not have to solve for X (or the length of the purple segment). Since the yellow (Y) and purple (P) top segments add to 5, and the Y & P segments also correspond to places on the bottom line for Y and on the second line for P, we know that together they will contribute an additional 5 units. You do not ever need to know what distance X is because you only need to know Y+P.
I like converting geometry to algebra whenever I can: set the purple length to x, then the horizontals are 5, x, 7, and 12-x, adding up to 24 (then 2 times 9 for the verticals), giving 42. If the x didn't "fall out", then you'd know the problem really is impossible!
Just draw a big rectangle of vertical side 9 and horizontal side 5 + 7, then it's quite easy to see that its sides can be composited from the sides of the original shape.
Start at the bottom left and consider each segment as a vector going clockwise. the short unknown segment will appear first as a negative vector and in the 7 unit segment as a positive vector. When you return to the stating point all of the vectors must equal zero. Use only the positive vectors for the horizontal dimension.
I looked at the horizontal piece between the 5 and the 7 (that was purple in the video) -- and just made an assumption about the length. The answer is independent of the assumption (try 0, 1, 2). Assume that it is 1, and then it is fairly simple to assign the correct length to the bottom segment and add them all up to get the correct answer.
I always say that is a good way to find the answer, good enough for school work. But one needs to be careful assuming a solution exists--some textbook problems are just wrong! Thanks for your support! (members see videos early)
I solved it a different way: if you move the yellow and green sides shown in 2:01 to the left (thereby shrinking the center and bottom rectangles) you end up with three rectangles: the top one with horizontal size 5, the center one with length 0 (I know, this may make no sense), and the bottom one with size 7. Perimter = 5 + 5 + 0 + 0 + 7 + 7 plus the verticals (both 9) = 42
An easier way is to look at the limits. If you move the green line all the way to the left you get two rectangles with top and bottom sides equal to 24. If you move the green line to the right do it meets the blue line you get horizontal sides of 5, 7 and 5+7. All of which add up to 42, so it doesn’t matter where the vertical green line is…
Isnt there an overlap between the 2 horizontal lines with length 5 and 7? The total length of the horizontal line (top or base) should be 5+7-Purple Line(which is unknown) Total Perimeter comes to be 42-(2*length of purple line)
Thank you! I thought I was crazy (or much dumber than I imagined) after reading all these “ah ha!” comments reiterating the same flawed logic. Or, is there an alternative definition of perimeter used here?
@@housercj1 Really weird there are no comments pointing this out. Looks like an obvious error in the video. Would be glad if someone can explain if I am wrong here :)
@@bt5289 I think you're pretty much spot on. While the concept is good, the execution here is incorrect. The perimeter is dependent upon the distance of the upper interior horizontal. If we knew the total overall length (like with the left vertical) or the distance of the upper interior, it would be a different story. Basically, it looks like many have overlooked the fact that the "7" incorporates the unknown distance, the proportion of which is indeterminable, and have transposed a sign in the various calculations. Ignoring the verticals, and thinking of it like a number line, the bottom length is L=5+(-x)+7 or L=12-x. Thus the total of the horizontals is L(t)=12-x+12-x, simplified to L(t)=24-2x. While we can use negatives to extrapolate certain lengths in these problems, perimeter itself is the sum of absolutes not the net of positives and negatives. Further, since the diagram is not actually drawn to scale, what would happen if the unknown upper leg was actually positive (i.e. extended to the right)?. Then L would be L=5+x+7. Unless x=0, L cannot be 12 and L(t) cannot be 24. Again, unless this video uses some sort of specialized definition of perimeter that I am unfamiliar with, it seems to be in error. I was seriously debating whether this was one of those psychological experiments to see if a person is courageous enough to point out the group is obviously wrong ;).
The perimeter is independent of the purple line. You can satisfy yourself that this is so by considering the two extreme situations where purple = 0 and, alternatively, purple = 5. In both cases the perimeter remains 42. The calculation is 18 [for the verticals] + 5 + purple + 7 + (7 + 5 - purple) = 42. That last bit is the length of the bottom horizontal.
@@mwatney9775 it cannot be. go to 3:17. Let's use his colors for the variables. given 5=y+p which means y=5-p. but the 7 **includes** this purple part! If it didn't, then the video would be correct. So, then L=y+p+(7-p) or L=y+p+7-p. From here, either substitute 5 for y+p or cancel the p's. Substituting gives us L=5+7-p, or L=12-p. If we cancel the p's, then L=y+7. substituting, then L=5-p+7 and we're back again to L=12-p. As a practical example: if I told you to take 5 steps forward (yellow +purple), then 1 back (purple), and then 7 forward again (orange), you'd end up a total of 11 steps from where you started and taken a total of 5+1+7, or 13 steps. Same process, if I told you to take 5 forward (y+p), 4 back (p), and 7 forward (o) again then your net distance traveled would be 8, but you'd have taken 5+4+7=16 total steps. Total steps is the perimeter. In this case, there is no way to determine the actual value of the purple part and hence no way to determine total steps.
The faster way to solve this is: 1) Assume there is a solution. 2) Then the length of that inset bit must be irrelevant to the solution, since we're not given enough information to determine how large it is. 3) Reduce the depth of the inset until it is arbitrarily small, then the bottom and "top" will each equal 12. Note that you can also reduce the height of the top bit and the inset to be arbitrarily close to zero, in which case you have a rectangle with sides of 9 and 12. The perimeter is left as an exercise for the student. 8-)
@2:35 “the horizontal lengths will magically work out in a similar fashion” No-there’s nothing “magic” about it. It’s just basic math. When people refer to things like this as “magic”, it causes people with lesser mathematical skills to actually think there’s deception or trickery involved, and to therefore doubt both the legitimacy of the answer and the integrity of the person presenting the answer. We have enough people in the world now whose knee-jerk response to anything they’re personally incapable of comprehending is to declare it “disinformation”, without also having people who are presenting factual truth also undermining their presentation by calling it “magic”.
Basic math is rather magical, though. I have always been enthralled by the way the numbers magically fit into each other. For instance, 1+4 = 2+3 = 2+2+1 = 5 or 2×30 = 6×10 = 4×15 = 5×12 = 2×2×3×5 = 60. It is not complicated. Simple things can be magical (like a sunset, an evening hanging out with friends, playing a game with one's parents, rain falling on a tin roof, etc) without being complicated.
But people who don’t understand math (or “magic”, in the sense of magicians performing show business tricks) equate “magic” with dishonest deception. By calling math “magic”, you are reinforcing that misplaced distrust.
@@verkuilb When someone tells you that an evening was magical, do you assume that they were scammed? When someone points out that flipping a switch on the wall "magically" turns on all the lights in a room, do you believe that they were deceived? He said that something magical occurred. He did not claim that math is magic (indescribable). He stated that something wondrous happened.
@@1st2nd2 I don’t-but I’m not talking about me or you. I’m talking about all the billions of simple-minded people in this world, who do think that math and science are literal, deceptive ‘magic’. We need to take affirmative steps to stop reinforcing their delusions.
@@verkuilb Do these hypothetical people believe that when they are told that an evening was magical, there was deception involved? I don't know anyone who believes that something happening "magically" involves deception. I have worked many years in retail and encountered many people you might label as "simple-minded" (I would not, btw.), and none of them would think that. Please be careful in labeling people as simple-minded due to believing that people are deceptive. I know plenty of relatively intelligent individuals who have been taken in by certain theories. That does not make them simple-minded.
What does the trick in these kinds of questions is try to change free variables and see if the problem is really impossible, like what happens to the shape and perimeter. In this case, increasing the length of the notch would "pull back" the length of the bottom, since the values of 5 and 7 are fixed, and that immediately tipped me up to the relationship between bottom and notch, which is key to the answer.
Again very nice one, but it could be solved in a much simpler way: once you assume all angles are 90 degree, you’ll realize you can “stretch” the figure to the right compressing the purple segments extractly as much as the red one to the point the purple is gone. Then you realize that you can compress similarly the vertical line green as much as you expand the yellow one. And so on until you realize you are bavk to a simple rectangle of sides 9 and 12 (7+5), which - guess what - has indeed perimeter 42…
Xs dont cancel. You cant solve for X. You are ignoring the middle line of X. Top Line = (5+7-X) Middle Line = X Bottom Line = (5+7-X) The real problem is 9+9+(5+7-X)+X+(5+7-X) = y [I added in parenthesis to make it easy to read] 42-X+X-X = Y This is your final answer 42-X=Y For this guy to get y = 42 then x = 0 which cant be true by looking at the picture. You have to know the value of X as X does not equal 0. The entire perimeter cannot be solved util you find the value of X which you cant do. You can only get 42-X=Y
@@common_c3nts you can get to the same solution assuming that X=2.5. The top line is NOT 5+7-X... the top line is just 5. Its labelled. Middle Line = 2.5 Bottom Line = 5+7-2.5 = 9.5 Thus you get, 9 + 5 + (9) + 2.5 + 7 + 9.5 = 42 where each term is (in order): the first vertical, the top segment, the sum of the right vertical segments, the middle line, & the bottom line. Your mistake is in thinking that the yellow segment on top equals 5, when it actually equals 5-X.
This is a very complicated explanation of a relatively simple solution once you get you head around it. Split the shape into it individual rectangles, note there are two of every individual line so add the numbers together and multiply the result by 2.
I put a big square 9x10 around the perimeter, and noted that the only extra length was the two short horizontal lengths in the wee notch, which had to be 2 long So the perimeter had to be equal to the hill perimeter square, 2(9x10) +4. To complete the question you need to give lengths to the sides of the rectangle on the other side of the equation; 6x7.
Left vertical = sum of right verticals. That’s 9+9=18. Now let shortest horizontal =x. Sum of all horizontals is: 5+x+7+(5+7-x)=24. Now perimeter is 18+24=42
Think of the purple section as a sliding section. If the purple section is 0, then the problem is easy as the vertical sides are 9 and the horizontal sections 12 (5 +7). If you slide the bottom two rectangles to the left, then the purple segment gets a variable length. The bottom orange / yellow line segment, however, will lose the exact length of the purple segment at the same time. So whatever length you have to add for the purple segment you end up subtracting from the bottom segment (the lengths cancel each other out) so the total perimeter remains the same (42).
I thought "you can just slide the thing back and forth", then realized that sliding takes away the same amount that it adds. So i could just imagine it was slid all the way to the left
Great for working out how much fencing you need, fence posts not so much. Get at least an extra panel (Maybe 2) to allow for the ones you will need to cut for the odd sizes.
The three vertical lines of unknown length adding to 9 is pretty obvious for a lot of people, I think. The way I figured out the other part was to name the sides a through g starting at the top and working counter clockwise. I then quickly saw that c = 5-g+7 so c+g = 5+7=12. Now you have everything you need. Cute puzzle.
This is easily solved by normal algebra. Set the unknown sides to a, b, c, d ,e, f, where "a" is the unknown part of the "5" side (white). Then: a+ c = 5 a+7 =f b+d+e=9 Perimeter = 21 +b+c+d+e+f =42 after substituting in. All the unknowns cancel out.
The three edges on the right-hand side of the shape add up to 9 (the same as the unbroken left-hand side). 9 + 9 = 18. Then, if we consider the overlap between the sides of length 5 and and length 7 as being equal to some unknown value x, that means that the bottom edge of the shape has length (12 - x). So the total of the 4 horizontal edges of the shape is 5 + x + 7 + (12 - x) = 24. Adding the 18 from the vertical edges gives 24 + 18 = 42. And that's the answer! (Presh voice 😄)
I think Presh may have missed the fact that the question's diagram was not just a visual demonstration that the diagram is rectilinear but that it was stating that the perimeter of the first rectilinear shape on LHS is the same as the perimeter of the rectilinear shape on the RHS. Let p1 be the perimeter of the LHS and p2 be the perimeter of the RHS. Let X and Y be the lengths of the sides of the rectilinear shape on the RHS. Then p2 = 2x+2y. For simplicity, choose one side of the shape (Y) on the RHS to be length 9 such that it matches the LHS. Then p2 = 2x+18. Let i, j, and k be the vertical unknown lengths on the LHS in descending order. Then (i+j+k) = 9. Let z be the horizontal unknown length between i and j. Then 5+7-x = z. Solve for x, x = 12-z. Then p1 = 9+5+7+(i+j+k)+x+z. Simplify p1 = (i+j+k)+x+z+21. By the question p1 = p2. Then i+j+k+x+z+21 = 2x+18. Solve for x, x = (i+j+k)+z+3. Then 12-z = (i+j+k)+z+3. Solve for z, z = (9-(i+j+k))/2. Substitute (i+j+k), z = (9 - 9)/2. Simplify z = 0. Then x = 12-0. Simplify x = 12. Since p1 = (i+j+k) + x + z + 21. Solve p1 = (9) + (0) + (12) + 21 Simplify p1 = 42. We arrive the same solution but while Presh's method works without the additional constraint of the question, discarding it leaves potential information on the table. Should a follow-up question have asked about the unknown length we called z he would not be able to answer it without adjusting methodology or acquiring additional information. This is evidenced by the fact that while he could demonstrate i , j, and k could vary without consequence to the perimeter, the same could not be done for z To be fair to Presh this question was presented poorly by whoever wrote it. It is improper to use an equal sign like shown in the image of the question without sufficient context denoting what is being equivocated or information can be misconstrued. While I am technically inferring that it is the perimeters of the 2 shapes being compared with the equals sign as much as Presh might have. These problems are generally designed to give no more information than what is required for solving it, so we must assume that the addition of the RHS of the equation is not redundant and is adding new, necessary information. A slight rephrasing of the question would make the problem much less confusing while maintaining intended difficulty. An addition mentioning that the diagrams are not necessarily drawn to scale, while not required, could point some students in the right direction if they are unsure where to start.
All the vertical lines on the right hand side of the figure add up to 9. If the horizontal distance between the two vertical lines on the right hand side of the figure is defined as x, then the perimeter is 5 + 9 + 5 + x + 7 + (7-x) + 9 = 42.
Since we are given that the vertical and horizontal perimeter sections can be easily separated, then the total is then the sum of the horizontal and vertical sections of the figure can be found separately and then summed. Let P be the total perimeter--- ----which is the sum of the vertical and horizontal sections. Pv =9+9 = 18, and Ph=5+7-x where x is the unknown horizontal section. The answer then is: Pv+Ph= 18+5+x+7+5+7-x = 42.
So lets add so notation. The line under 5 and above 7 is x so the last line is 12-x. Then we note vertical lines with a,b,c. A+b+c= 9 Then we have 2 other unknow lengths of x and 12-x that when added will be 12 So the total is 9*2+12*2=42
I figured the bottom was the two horizontals added together minus their overlap so 5 + 7 - (7 - 5) so the bottom is 10 and the middle unknown is 2 so when combined with the obvious sides you get 9 + 9 + 5 + 7 + 10 + 2 = 42.
1:50 When you have to calculate the perimeter you can do changes, that influence the area but not the perimeter. E.g. like shifting the line marked with 7. Just shift it to the right until it´s left end is under the right end of the line marked with 5. (The next step is to make a rectangle of it, if you need it). So you have 9 + 5 + 7 for half of the perimeter.
I struggled with this one for a bit because we don't know the length of the overlap, and then I hit on an idea - what if the length of the overlap doesn't matter? To test it it, I assumed it was 2 and calculate the perimeter as 42. Then I assumed it was 3, and got the same result. So then I called the overlap x, and set up the equation, and found that x cancels out.
The answer is even easier, if you assume the length of the bottom part is 5+7 then you'll be over counting since 5 and 7 have overlap. so just don't count that part of the perimeter (the purple overhang part) and it will all balance out.
Alternative visualisation is to imagine you walk along the perimeter until you reach your starting point again, let's say in counterclockwise direction. Clearly, we have to walk as far up as we walk down, and also as far left as we walk right. Since we know all edges that we walk along downwards and left, and all edges we traverse upwards or right are unknown, the perimeter is twice the sum of all known edges.
Just giving it a shot before the vid: labelled each unknown side length a, b, c, d, e starting from top going clockwise around the shape. from known sides we get that: a + c + d = 9 5 + 7 - b = e ----> 5 + 7 = b + e so combining these we get that (a + c + d) + (b + e) = (9) + (5 + 7) -----> a + b + c + d + e = 21 so all the unknown sides add to 21, and the known sides also add to 21 (9 + 5 + 7 = 21) so 21 + 21 = 42 is the answer
Move the green line in vertical alignment with the blue line, moving the 7 line, and connected lines, with it (so, as the horizontal line between the blue and green reduces, the baseline increases)... then... move the 7 line, and connected lines, upwards to align with the 5 line, forming a rectangle, 9 high by (5 + 7) wide... ... with perimeter (9 + 5 + 7) x 2 = 42
I was too caught up in finding the values of the missing sides to notice that you only need to find the sum of them. This is a nice way of thinking about perimeters.
If you think of it as a belt going around tiny little pulleys, it's easy to visualize stretching this out horizontally while maintaining the perimeter. Then you can pop out that last corner to complete a 9x12 rectangle. Hardly any math required.
Your way was so much simpler than mine. I calculated the two areas (9x7) and (9x5), found them difference which would be the area overlapping (18cm square) and then found it’s perimeter (9x2). So the length of the red side was 2cm. Worked out the horizontal lengths from there and worked out the vertical length as you did. Got the same answer but took way longer. Thanks for the video.
A much clearer approach I think is to label the unknown sides and use substitution. Let a = the unknown side perpendicular to side of length 5 Let b = the unknown side perpendicular to a Let c = the unknown side perpendicular to b Let d = the unknown side perpendicular to side of length 7 Let e = the unknown side comprising the base, i.e. the line segment perpendicular to d and side of length 9 Now the definition of perimeter yields 9 + 5 + 7 + a + b + c + d + e => 21 + a + b + c + d + e We know from the colorful coding that a + c + d = 9 so 21 + 9 + b + e => 30 + b + e e = 5 + 7 - b or, e + b = 12 => 30 + 12 => 42
If there is a single answer, which the question asking for a single answer would have you believe, then the wiggle to the left can just as well be zero size as any other size. In which case both top and bottom are 12, and left and right are 9, hence 42. If you want certainty you can observe that whatever length we increase the wiggle to, an equal amount is added to the wiggle, and subtracted from the bottom.
Looks bad until you start writing down what you know. Perimeter is made up of horizontal components and vertical components. Getting the total vertical is easy, there's no overlap. Pv = 9 + 9 Then write down what you know about the horizontal, with the two unknown segments labeled x (short) and y (bottom). You can express y in terms of x. y = 5 - x + 7 Then write down what the total horizontal component of the perimeter is. Ph = y + 7 + x + 5 Substitute in your equation for y, so the horizontal perimeter is only in terms of x. Ph = 5 - x + 7 + 7 + x + 5 Viola, x cancels out and you're done. Well, add the horizontal and vertical components...
I wonder how many stumped students just put 42 to be a smartass and accidentally got it right
It's sure a weird shaped towel though.
@@lidarman2 Towel? I thought it was my garden!
i guessed 42 havent watched video yet
I quickly figured out 42. Problem was easy. The "solution" as stated in the video, however, was too confusing for me.
@lidarman2 Ah - a man with a towel! I see you came prepared for anything.
I really thought it was impossible till I did the math. Obviously vertical is 9+9. Then I said the small horizontal is x and the bottom is 12-x. Put it all together and the x's drop out. And the solution is the answer to life, the universe, and everything!
Exactly, and now we finally know what the question was!
First time I can relate to someone in these comments of these videos. You actually make sense.
I solved it the same way .
At first read I thought they were asking for the area 😂, I read the question and though, oh, that's easy: consider the inner horizontal segment be 0 and that's it.
Hitchhikers guide to the galaxy reference
And my dumbass was trying to calculate area this whole time.
Same idk why i wanted to find the area so badly
Yeah, I thought that must be the trick. But it was sort of easier this whole time.
Mee tooooo
Thank god I wasn’t the only one😭
Same😅
Nerdy point here: 42 is not the answer to Life, the Universe, and Everything (nor is it the answer to the Meaning Of Life which some people posit). It is instead the answer to The Ultimate Question Of Life The Universe And Everything, which has not yet been asked. We don't know what the Ultimate Question even is.
Well we do now. It’s this question.
Wasn't the question discovered to be "What do you get if you multiply six by nine?" in one of the books?
@@zuggo Technically sure that’s definitely what the book says, but that’s lame and I don’t like it.
@@zuggo The assumption was that as Arthur was a human from Earth, he must have The Question buried in his brain laying dormant, and they just needed to tease it out. However the result was not helpful, and it was decided that was probably not The Ultimate Question after all.
@@GuanoLad ah thanks, I haven’t read these books in a decade I should again
It's actually quite simple if you look at it algebraically. On the left side, we already have 9, and the right side has no overlaps, and all angles are 90 degrees, meaning that the three segments must add up to 9 to match the left side. So you don't even need to know the values of each individual line. Then the top we do have an overlap, so we can't do the same thing. However, we can give the short segment a value of x. So, the bottom line will be 5+7-x, because x is the overlap. And for the top, because we're checking the perimeter, we have to include all three, so it's 5+7+x. So, all in all it's 9+9+5+7+x+5+7-x. Add up all the numbers and you have 42+x-x. Well, now the x's cancel each other out, leaving only 42 for the answer, so you don't even need to know the value of x.
very underrated comment 💯
We don't know the purple line length!
@@soundsoflife9549 We don't need to know it.
@@soundsoflife9549 That's why he called it x and just ignored it after it canceled out in the calculation.
This is how I did it.
Call the short horizontal side x. It follows that all horizontal lines contribute 5 + x +7 + (5-x) + 7 = 5 + 7 + 5 +7. All vertical lines contribute 9 + 9. The total is 42
That's how I solved it too.
Better than the video
No, you would call it x. X and x are two different variables.
@@robertveith6383 Cringe
@@robertveith6383 right big X stands for Xehanort and we wanna avoid that
imagine if it said "not drawn to scale"
Which is usually the case in these problems
You gotta assume that
doesn't make a difference
They aren't drawn to scale. The only thing why it works is because the lines are straight. Solving this on your own gives you alot better of an understanding of the problem
It wouldn't matter? The solution shown does not depend on the picture being to scale
Some of us are colour blind. Can we go back to using letters to label diagrams please?
yea. also I don't carry colored pencils with me, so when doing it on paper myself then I'm just going to use letters.
Not that i care much about the colorblind, but it also looks more childish and less math-y. We're making formulas ans algebra, just use math instead of colors like its for babies
There are four horizontal edges: one length 5, one length 7, the one between them (call it A) and the one at the bottom (call it B). If you increase length of B, you decrease the length of A by the same amount, so the perimeter stays the same. If you continue this until A has length zero, the bottom edge has length 12.
This is exactly how I worked it out.
@@neilpadfield Same here. I changed the lengths and it worked out the same every time.
I had to watch that twice to get his point about the horizontal lengths. I think this way would be easier:
Call the length he marked in purple, x.
The total horizontal lengths are the top line (5), plus the purple line (x) plus the orange line (7), plus the bottom line (5 + 7 - x).
5 + x + 7 + (5 + 7 - x) = 24. (The x's cancel).
24 + the vertical lines (18) = 42.
I feel like the yellow-purple-orange line could use a bit more explanation, I got it after staring for a few mins but as presented it feels like a jump in logic.
We know that yellow + purple = 5 and orange = 7. The bottom line is equal to yellow + orange. Then we need to add the small purple part that “goes” inwards. We add 5 + 7 a second time and without knowing the exact numbers we are able to determine the perimeter of the shape.
same. i somehow forgot that the purple line exists 🤦🏻♂️
so lets say the purple line is length x. The bottom line will be 5 + 7 - x, cause that length x is included in both the 5 and the 7, so it's counted twice and we only need it one time. So now, if you go around and add all the lengths, you get: 9 + 5 + 9 + x + 7 + (5 + 7 - x), where the second 9 is the sum of the right hand vertical lengths. You see that in that result, both x values are canceling each other, and you get the remaining 9 + 5 + 9 +7 + 5 + 7 = 42
It might have been better to use a different color than yellow, since that had already been used for the vertical lines. By making it a different color, it becomes clearer that the perimeter consists of two sections each of six different colors, and therefore the sum is 2x(9+5+7).
It could help to use some Algebra.
x is the part of the side of length 5 (or "line A" to simplify the rest of this explanation) that's _not_ included in the side of length 7 (or "line B").
y is the part of A that _is_ included in B.
z is the part of B that's not included in A.
The bottom line (line C) is the sum of x, y, and z. There's also an additional y from where the shape moves left before moving down and right again.
x + y = 5 (line A)
y + z = 7 (line B)
x + y + z = ? (line C)
Perimeter: 9 + 9 + 5 + y + 7 + (x + y + z)
(Parentheses around the line C part).
Substitute the first y with (5 - x). Substitute the second y with (7 - z).
9 + 9 + 5 + (5 - x) + 7 + x + (7 - z) + z
x and z cancel.
9 + 9 + 5 + 5 + 7 + 7 = 42
I thought the question was too easy, and thus questioned myself until I got 42, then I no longer questioned myself as to the answer.
You turned 42?
@@SyedAhmedJaved 42
42 likes
@@SyedAhmedJavedhahaha
@@ILoveLuhaidan ﷽
وعلَیکم السلام ورحمة اللّٰه وبرکاته
From Pakistan
Just visually stretch out the bottom horizontal line until the indentation goes away: 5+7=5+7. The right-side verticals are pretty obvious: whatever they are = 9. 12+12+9+9=42
Yes, that works for me.
Yes, that's essentiallyhow I did it too. My visualization was to slide the top left part (sides labeled 9 and 5) leftward while keeping the unlabeled bottom and right sides in place. While doing this, all but two sides keep their lengths. The purple side gets smaller, while the bottom side gets larger by the same amount. Continue until the purple side has reached length 0, so that the overall shape is that of an L. Or a rectangle with the top right corner bitten off. Biting off a rectangular corner from a rectangle does not change the perimeter, which is quite obvious if you have drawn the image. So the solution is simply the perimeter of the full rectangle, which has horizontal sides of length 5+7 and vertical sides of length 9. So the overall perimeter is 2 x (5+7 + 9) = 42.
You're showing that the limit as x->0 of the perimeter of this shape is 42. What you haven't shown is that the perimeter is indifferent to the value of x.
@@rickdesperNo. You're taking the purple line x out and putting it in the bottom edge. This doesn't alter the perimeter and only simplifies the shape.
The equals sign and normal rectangle threw me off so much because it made me think I was somehow supposed to be manipulating the shape into a rectangle first and then finding the perimeter, as soon as I got that idea out of my head it was much easier.
4:12 "Life, the universe, and everything" isn't a question, so can't even have an answer. 42 is the answer to the ultimate question about life, the universe, and everything.
You're implicitly telling us you haven't read The Hitchhiker's Guide to the Galaxy.
@@fopdoodler9427 How do you figure that? Not only have I read it, multiple times, in multiple languages, but apparently you haven't read it *and* are too dim to even understand basic English. Unlike Mr Adams, who's a master of it. Your comment is almost, but not quite, completely unlike a sane one, in much the same way that mine wasn't.
@@fopdoodler9427 From your comment, it rather sounds like you're the one who hasn't read it, or if you have read it didn't understand what you read.
@Grizzly01-vr4pn I did read it, my dear sir. And I just checked online copies to see where it's written and actually found it again, that's why I gave a page number. It won't be the same page number for every copy, but page numbers are an indicator.
@@fopdoodler9427 page number written in invisible ink, is it?
Although it's nice to get to an answer by only ADDING all the bits, I think a more satisfying way is to say that the UPPER horizontal lines add to 5+7+x (where x is the purple bit), and the base line comes to 5+7-x (ie, this time x SUBTRACTS). Then, when we add all the bits together, the x will cancel out (x-x), and so we get to a numeric result. I think this response helps us to better understand intuitively WHY we can get to a result even when we don't know how long x is.
Might have been more interesting at the end to slide the 7 length horizontally (along with the adjoining sides), and show that the perimeter remains 42 with that translation as well.
The part I don't understand is what that other rectangle on the right is, and what does it mean they're equal?
I think it means they are both rectilinear by definition, i.e contained by, consisting of, or moving in a straight line or lines, despite the left polygon being a composite of different shapes.
that was unnecessary BS to look like they know what rectilinear means
I guess this was the teacher's reminder of the solution path the students were presented in class. It may have made sense to the students, but without such context it's too obscure and in fact seriously misleading.
Here is my solution path, which retroactively made sense of the hint:
Slide the top left part of the shape (sides labeled 9 and 5) leftward while keeping the unlabeled bottom and right sides in place. While doing this, all but two sides keep their lengths. During this process, the side painted purple in the video gets smaller, while the bottom side gets larger by the same amount. Continue until the purple side has reached length 0, so that the overall shape is that of an L. Or a rectangle with the top right corner bitten off. Biting off a rectangular corner from a rectangle does not change the perimeter, which is quite obvious if you have drawn the image.
So the solution is the perimeter of the resulting full rectangle, which has horizontal sides of length 5+7 and vertical sides of length 9. So the overall perimeter is 2 x (5+7 + 9) = 42. This resulting full rectangle is probably what the rectangle on the right hand side of the equation was supposed to be.
@@IKVKronosOne In this video "rectilinear" is being used to refer to a shape where all angles are right (a mix of 90° and 270° angles, as applicable), since polygons are already defined to be made of straight line( segment)s. Or equivalently, a shape where any two sides are either perpendicular or parallel to one another.
The rectangle could be a blank to write the answer in. It is a little bolder than expected of an answer box.
I'm used to so much "Not drawn to scale" that honestly i just assumed it wasn't drawn to scale. honestly i think problems like these should always require mentioning whether or not they're drawn to scale.
Don't you assume they're not drawn to scale unless specified otherwise?
. . . are they ever specified as drawn to scale unless you're exploring something empirical and using rulers or whatever to demonstrate the rule?
@@awfuldynne I was taught to *always* assume shapes are not drawn to scale, a problem like this would not be allowed as is
it doesn't matter herr actually, the relevant information was given in the question
I don't think it necessarily is drawn to scale. I'm pretty sure you can assign any values you want that are less than 5 to the horizontal unknown and assign any values to the three vertical unknowns, as long as they add up to 9, and it will always be 42.
It doesn't need to be drawn to scale, the result comes frkm a logical analysis, you don't need to measure anything.
Regarding the last part of video where you extend the vertical green line: That's how I solved it.
1. Extend the green line up and down until it achieves a length of 9, forming a standard rectangle with dimensions 9 by x.
2. Double the two edges that stick out as these are "rectangles" with width 0. So this part of the perimeter is 14 and 2(5 - x)
3. The perimeter of the large rectangle is 18 + 2x
4. Total P = 14 + 2(5 - x) + 18 + 2x = 14 + 10 - 2x + 18 + 2x = 14 + 10 + 18 = 42
I got confused at around 3:15 when I didn't realise the 2 * "Yellow + Orange = 5+7" contains the two extra "Purples" in the middle, maybe this could have been made more obvious at this point by pointing it out.
Ditto. It is implied that the purple bit is not Zero units, and the sketch implies it must be less than 7
[Edit: I think...]
We never actually work out the value of the purple segment. We only know that the bottom edge plus the purple segment = 7+5
Yeah, I got that eventually, it just took a while for me to click that the two purple segments in the middle are covered by that.
I.E. that the top part is "Yellow + Orange" (and not Yellow + Orange + Purple), the bottom part is also "Yellow + Orange".
My argument is that it would have been better if Presh specifically pointed this out when he was rearranging the colored segments (if he did, I completely missed that).
@@caoryn It's because he very clearly says that we know that yellow+purple+orange "must equal 5+7," which, of course, they don't. It's just that the unknown length of purple cancels out when you add up all the sides.
@@gabbleratchet1890 Yeah, and it would be much more obvious if he'd spent the five seconds to specifically mention that. 🙂
0:12 you haven't even asked the question yet and I'm sitting here with the answer to life, the universe, and everything. Will you mention that? Edit: YOU WILL!
I work in a machine shop and have to do a lot of this with different parts because on the technical drawing not all dimensions are directly given some are often implied.
The only thing I would say is I had the a+b+c = 9
But I saw and solved the horizontal portion as 5 + 7 which were given but I broke it down in (7-5) and (5-(7-5)) for the portions where there is overlap on the 5 and 7 horizontal lines so that it was 9+5+a+(7-5)+b+7+c +7+(5-(7-5))
Where a+b+c =9 or 9+9+7+7+5+(5-(7-5))+(7-5) which once I saw the video I realized how it really could be written more similar how he did as 9+9+7+7+5+5 because all I did was break the second 5 down to solve for each of those individual lengths
my silly ass thought i was supposed to find the numbers for all of the sides one at a time but you oculd of just, skipped them entirely? this video enlightened me lmao
Isn’t it 5 unknown sides at first?
It sure is!
it turns out, we don't care! we can still figure the answer out without ever finding out the lengths of the unknown sides.
@@raffimolero64More to the point, the lengths of the unknown sides don't matter.
Yes.
3 of the 5 unknown sides add up 9.
Thank you Professor Layton and the Curious Village for having this exact puzzle in it
Lets say we moved the green line to the right lengthening the bottom line. The bottom line lengthens the same amount as the line at the top of the green shorten so we could make it so that the 7 starts where the 5 ends. This essentially would make it about calculating the perimeter of a rectangle.
Indeed. That's how I solved it, and it's a much simpler way.
To be more exact, one would need to move not only the green line to the right, but the other vertical lines as well in order to end up with a rectangle. First move both the green and the bottommost vertical line to the right by the same amount until the green line joins the topmost vertical line to become a single edge of the shape. Then move that edge to the right until the shape becomes a rectangle.
There are multiple simple ways to solve this problem and any solution will seem easy to some people but difficult to others.
Xs dont cancel. You make no sense. The guy in the video trolled all of you. You cant solve for X. You are ignoring the middle line of X.
Top Line = (5+7-X)
Middle Line = X
Bottom Line = (5+7-X)
The real problem is 9+9+(5+7-X)+X+(5+7-X) = y [I added in parenthesis to make it easy to read]
42-X+X-X = Y
This is your final answer 42-X=Y
For this guy to get y = 42 then x = 0 which cant be true by looking at the picture. You have to know the value of X as X does not equal 0.
The entire perimeter cannot be solved util you find the value of X which you cant do. You can only get 42-X=Y
@@common_c3nts What are you talking about? There are not three horizontal edges, but four. From top to bottom, they have length 5, X, 7, and 5+7-X. The sum of these lengths is 24. No dependency on X left. The lengths of the vertical edges sum to 18, giving a total perimeter of 42.
@@alessandropizzotti932 It can't be solved, there's 2 variables that depend upon each other with no relation to any of the rest of the figure.
A very intuitive way to present it. I ended up doing a lot of extra work making equations for everything I didn't know, adding it all up, then substituting until I got the answer.
Xs dont cancel. The guy in the video trolled all of you. You cant solve for X. You are ignoring the middle line of X.
Top Line = (5+7-X)
Middle Line = X
Bottom Line = (5+7-X)
The real problem is 9+9+(5+7-X)+X+(5+7-X) = y [I added in parenthesis to make it easy to read]
42-X+X-X = Y
This is your final answer 42-X=Y
For this guy to get y = 42 then x = 0 which cant be true by looking at the picture. You have to know the value of X as X does not equal 0.
The entire perimeter cannot be solved util you find the value of X which you cant do. You can only get 42-X=Y
I didn't understand your second step, but it works for me if I do it algebraically.
His explanation lacked a little clarity. The trick is in that the horizontal distances are given from opposite ends. I had to do it algebraically myself to see how the center horizontal distance cancelled.
Yeah same, except I did it algebraically first so his explanation felt like it was skipping steps.
Perform a geometric transformation that does not affect the perimeter: shift the vert. green line, the hor. white 7-line and the vert. yellow line to the left until the green line meets the vert. white 9-line. The hor. purple line will grow and the hor. bottom line will shrink, and both changes will compensate each other. Then the perimeter of the shape is clearly: 2*5+2*7+2*9=2*21=42
I knew intuitively that it was 42-but your color coding was a great way to make it clearer for people who aren’t as nimble in visualizing such things on their own (such as the supposed “baffled parents”). Well done!!
I let the purple length equal x.
The horizontal segments are now 5, x, 7, and the bottom is 5+7-x, as we have to remove the overlap.
The sum of the horizontal segments is 5+x+7+5+7-x. That sums to 24 as the x values cancel each other.
Now add 18 for the vertical segments to get to the solution of 42.
Imagine answering 42 for the memes and it actually being right...
I was so convinced that this was so much more complicated than it was that when I did the step as you described I couldn't believe it was that simple. The curse of knowledge I suppose.
I’m surprised in the final part of the video, when you changed the shape by moving the rectangles up and down (but maintained the same perimeter) you didn’t also demonstrate that you can also adjust the horizontal width of the shape (maintaining 5 and 7, but changing the amount those lengths overlap), and still have the same perimeter…
You can even make a rectangle out of it by setting green, blue and violet to zero, hence getting a rectangle of sizes 9 & 12, then getting 42 is trivial
Exactly, that is how I solved it in my head. There is the freedom to shift the horizontal 7 to the right, until it is easy to see the horizontal lines together must have length 2 x (5+7).
You can vary the purple line!
Isnt that because increasing the length of one decreases the length of the other?
@@Dr_mafario yes, that is why it works.
This was a Professor Layton 1 puzzle, and it was SO satisfying to realize it was actually solvable!
The horizontal perimeter is given by the sum of bottom + the sum of the 3 other horizontal segments. The bottom = 5 + 7 - purple; the 3 segments = 5 + 7 + purple. Together you get the horizontal perimeter to be 5+7-purple+5+7+purple. The purples cancel and you get 5+7+5+7=24. Add the 9x2 vertical and you get 42.
Don't the top and the bottom necessarily have to have the same length? So if the top is 5 + 7 - purple, the bottom must also be 5 + 7 - purple?
@@raifmt Right. If you think about making a rectangle, then of course the top and bottom must be equal. I guess it was easier for me to think about adding up all 4 horizontal segments. But your approach absolutely works too.
I could quickly see that all the unknown vertical sides must add up to 9.
It took me a long time to understand how to deal with unknown horizontals. Having read a lot of other people's solutions I have come up with my simplest way of dealing with horizontals:
- notice that shrinking the shorter unknown horizontal, and stretching the longer one by the same amount does not change the perimeter,
- shrink the shorter to zero (and stretch the longer)
- result is an L shape, where it easy to see that the bottom side is 7+5 = 12
Verticals add to 2 x 9 =18
Horizontals add to 2 x 12 = 24
18 + 24 = 42
One slight mental block I had was trying to keep the area the same, whereas that was not relevant.
Also the feeling that I could do this easily if I was 50 years younger.
Its dishonest if the question is written to assume the lines are drawn to scale and the user can use a measuring stick without explicitly indicating such in the problem statement. A lot of times, problems like these are purposely not drawn proportional to the given values.
There's no need to measure any of the sides in this problem.
@@Jeremo-FD Ya, you're right. I had initially thought that he was implying the horizontal yellow and purple were equal parts so 2.5 + 2.5 + 7 on the horizontals. When it actually looks like he was implying some theorem about rectumlinear proportionality with the sum of the lengths for 5 and 7. It makes more sense to think of the sum of the yellow and purple as 5 with the two unknowns not being important. Then of course the orange being 7 because of the rectumlinearness.
@@Commenter-f8bPresh's explanation certainly was lacking. He didn't specify *how* he chose the yellow and the purple part. So I see how you got confused here.
This problem is kind of nice, but as usual, there's no single explanation that is absolutely clear to everyone. This particular problem would have been a good opportunity to show different approaches. Some I could think of at the top of my head:
1) Compare edge lengths visually like in the video.
2) Name the edges and do it algebraically.
3) Change the shape in a "continuous" fashion that does not change the perimeter until you end up with a rectangle. E.g. take the 7-edge and its two adjacent vertical edges and move them to the right until the "purple" horizontal edge disappears, leaving with an L shape. Then move the 7-edge up until you have a rectangle.
4) Change the shape by repeatedly placing parts of it at different places until you have a rectangle. E.g. rotate the three "indent" edges by 180° so that they now form a "bulge" instead of an indent. Then get rid of the 270° angles by taking the the two incident edges and again rotate them by 180° so that they no longer point "inwards".
5) Follow the outline of the shape until you reach you starting point again and observe that for both the vertical and the horzizontal edges, the ones whose lengths we know are traversed in one direction, while all the others are traversed in the opposite direction.
(3) Making the “L” is easiest. The length of the purple inside horizontal segment is arbitrary, so just choose zero and make the “L”.
The perimeter of an “L” is same as the perimeter of the rectangle of same base and height, because the two “inside segments” are identical to the original upper right of the rectangle. Just kind of dented in.
Anybody with a modicum of geometric visualization ability will see it pretty fast. And it’s a very satisfying, elegant answer.
Really cool challenge. Thanks a lot!
A little bit of pendantry though:
0:56 a rectangle with a semicircle at two ends is not called a pill. In geometry pill is a three dimensional shape. Cut that shape in half and rotate it along a circle and you have a pill. (The shape is commonly used as a collision volume in virtual reality and for physics simulation btw. It's one of the four volumes that is easiest for a physics engine to handle - the other three are the cube, the sphere and the cylinder.)
The two dimensional shape shown in the video is usually called a stadium. I've never heard the name trackhouse bedore but it does fit.
Wikipedia would disagree with you.
> A stadium is a two-dimensional geometric shape constructed of a rectangle with semicircles at a pair of opposite sides.[1] The same shape is known also as a pill shape,[2] discorectangle,[3] obround,[4][5] or sausage body.[6]
While the 3d shape is called a capsule.
@@woobilicious. Oh, maybe I was wrong then. I only know this from working with 3D graphics.
But in that case, we should probably avoid using the "pill" name at all if it's ambigious. The most common name for the 2D variant is still "stadium" and the 3D one is also commonly known as a "capsule". Those two words are unambigious in geometry so prehaps we should stick to them.
Love a problem that looks difficult, but solved it in about a minute. Thanks for posting.
Original question had a composite shape = rectangle shape. Totally meaningless.
Not so. It helped me solve it visually. The composite rectilinear shape can be converted to the rectangle.
You move the 7 horizontal line to the top to meet the 5. All of the vertical lines are moved to the right side edge. Then the leftover short horizontal line must fill in the bottom edge to complete the rectangle.
Now you have a 9x12 rectangle.
I realise that but the diagram as it stands does not suggest this.
Got it right! I did mine a bit different as I broke the 7 up as the overhang part must be the difference between 7 and 5 so I ended up with the top horizontal line being (5+5+2+2) leaving the bottom line 5+5.
We arrived at the vertical lines values of 9 each the same so (9x2)+(5x2)+(5+5+2+2)=18+10+14= 42
The distributive property of addition is great!
Is it 42?
Yep!!
Consider the horizontal lines. Relocate the 7 to join onto the 5. This leaves the unknown short length. By inspection the right hand end of the 7 has moved right by the length of the short unknown so this unknown can fit onto the right hand end of the base, making the base 7 + 5. = 12. So we have two equal horizontal lengths of 12 to add to two vertical lengths yielding the equivalent perimeter.
How is this impossible, I did it in like 5 minutes.
1:51 Shift the green line right, to be under the blue, sliding the 7 line and yellow line the same distance. The bottom line is lengthened the same length as the middle horizontal that is eliminated. Easily visualized as 5+7+12 + 9+9
Professor of computer science here. While I appreciate showing how to solve this graphically. I think you could have been clearer that the reason this can be solved is because there is exactly one unknown variable. The overlap between the side of length 5 and the one of length 7 and that the perimeter can be calculated by adding that value and then subtracting it again. Thus eliminating the variable. A shape with extra variables would not be solvable. Writing this out algebraically might also have been a good idea and I suspect that would give students an understanding about how to solve this problem more generally.
Xs dont cancel. You make no sense. The guy in the video trolled all of you. You cant solve for X. You are ignoring the middle line of X.
Top Line = (5+7-X)
Middle Line = X
Bottom Line = (5+7-X)
The real problem is 9+9+(5+7-X)+X+(5+7-X) = y [I added in parenthesis to make it easy to read]
42-X+X-X = Y
This is your final answer 42-X=Y
For this guy to get y = 42 then x = 0 which cant be true by looking at the picture. You have to know the value of X as X does not equal 0.
The entire perimeter cannot be solved util you find the value of X which you cant do. You can only get 42-X=Y
I disagree, the verticals are not a problem as those are all right angles so the verticals on the right must add up to the vertical on the left. That's fine. However, we started out with 5 unknown sides based on the problem, 3 of which are verticals that must add up to the right hand side of 9, giving us 2 unknowns without the information to solve it. Even if we divide the bottom the way the video does, that still gives us a section that cannot be figured out with the information provided. You'll note that he didn't even try to lengthen or shrink that bottom segment.
The video errs in assuming that the bottom is 9 due to scale. That cannot be assumed unless it is stated to be the case.
@@common_c3nts incorrect. You do not have to solve for X (or the length of the purple segment). Since the yellow (Y) and purple (P) top segments add to 5, and the Y & P segments also correspond to places on the bottom line for Y and on the second line for P, we know that together they will contribute an additional 5 units. You do not ever need to know what distance X is because you only need to know Y+P.
We finally know why it took so long for the Robot to caculate the meaning of life, the universe and everything.
我们对这类题有一种称为“拍球法”的方法。
第一步：找一个起点，和一个方向(顺时针或者逆时针)
第二步：沿着方向走一圈，↑走的过程中，记录下每条线段的方向。↑→↓←
然后是结论：所有↑的线段总和=所有↓的线段总和；所有→的线段总和=所有←的线段总和
利用这两个结论，你可以很快算出类似结构图形的周长。会比平移线段的方法快很多。
Ok, that's actually pretty clever!
好吧，这实际上非常聪明!
I like converting geometry to algebra whenever I can: set the purple length to x, then the horizontals are 5, x, 7, and 12-x, adding up to 24 (then 2 times 9 for the verticals), giving 42. If the x didn't "fall out", then you'd know the problem really is impossible!
Just draw a big rectangle of vertical side 9 and horizontal side 5 + 7, then it's quite easy to see that its sides can be composited from the sides of the original shape.
Start at the bottom left and consider each segment as a vector going clockwise. the short unknown segment will appear first as a negative vector and in the 7 unit segment as a positive vector. When you return to the stating point all of the vectors must equal zero. Use only the positive vectors for the horizontal dimension.
I looked at the horizontal piece between the 5 and the 7 (that was purple in the video) -- and just made an assumption about the length. The answer is independent of the assumption (try 0, 1, 2). Assume that it is 1, and then it is fairly simple to assign the correct length to the bottom segment and add them all up to get the correct answer.
I always say that is a good way to find the answer, good enough for school work. But one needs to be careful assuming a solution exists--some textbook problems are just wrong! Thanks for your support! (members see videos early)
If we just assume as x, we will get 24 as sum of all horizontal lengths.
I solved it a different way: if you move the yellow and green sides shown in 2:01 to the left (thereby shrinking the center and bottom rectangles) you end up with three rectangles: the top one with horizontal size 5, the center one with length 0 (I know, this may make no sense), and the bottom one with size 7. Perimter = 5 + 5 + 0 + 0 + 7 + 7 plus the verticals (both 9) = 42
This is one of the lovely questions where, if you look at what you don't know, you get stuck, but if you just DO IT, it works.
An easier way is to look at the limits. If you move the green line all the way to the left you get two rectangles with top and bottom sides equal to 24. If you move the green line to the right do it meets the blue line you get horizontal sides of 5, 7 and 5+7. All of which add up to 42, so it doesn’t matter where the vertical green line is…
Isnt there an overlap between the 2 horizontal lines with length 5 and 7? The total length of the horizontal line (top or base) should be 5+7-Purple Line(which is unknown)
Total Perimeter comes to be 42-(2*length of purple line)
Thank you! I thought I was crazy (or much dumber than I imagined) after reading all these “ah ha!” comments reiterating the same flawed logic. Or, is there an alternative definition of perimeter used here?
@@housercj1 Really weird there are no comments pointing this out. Looks like an obvious error in the video. Would be glad if someone can explain if I am wrong here :)
@@bt5289 I think you're pretty much spot on. While the concept is good, the execution here is incorrect. The perimeter is dependent upon the distance of the upper interior horizontal. If we knew the total overall length (like with the left vertical) or the distance of the upper interior, it would be a different story.
Basically, it looks like many have overlooked the fact that the "7" incorporates the unknown distance, the proportion of which is indeterminable, and have transposed a sign in the various calculations. Ignoring the verticals, and thinking of it like a number line, the bottom length is L=5+(-x)+7 or L=12-x. Thus the total of the horizontals is L(t)=12-x+12-x, simplified to L(t)=24-2x. While we can use negatives to extrapolate certain lengths in these problems, perimeter itself is the sum of absolutes not the net of positives and negatives.
Further, since the diagram is not actually drawn to scale, what would happen if the unknown upper leg was actually positive (i.e. extended to the right)?. Then L would be L=5+x+7. Unless x=0, L cannot be 12 and L(t) cannot be 24. Again, unless this video uses some sort of specialized definition of perimeter that I am unfamiliar with, it seems to be in error. I was seriously debating whether this was one of those psychological experiments to see if a person is courageous enough to point out the group is obviously wrong ;).
The perimeter is independent of the purple line. You can satisfy yourself that this is so by considering the two extreme situations where purple = 0 and, alternatively, purple = 5. In both cases the perimeter remains 42. The calculation is 18 [for the verticals] + 5 + purple + 7 + (7 + 5 - purple) = 42. That last bit is the length of the bottom horizontal.
@@mwatney9775 it cannot be. go to 3:17. Let's use his colors for the variables. given 5=y+p which means y=5-p. but the 7 **includes** this purple part! If it didn't, then the video would be correct.
So, then L=y+p+(7-p) or L=y+p+7-p. From here, either substitute 5 for y+p or cancel the p's. Substituting gives us L=5+7-p, or L=12-p. If we cancel the p's, then L=y+7. substituting, then L=5-p+7 and we're back again to L=12-p.
As a practical example: if I told you to take 5 steps forward (yellow +purple), then 1 back (purple), and then 7 forward again (orange), you'd end up a total of 11 steps from where you started and taken a total of 5+1+7, or 13 steps. Same process, if I told you to take 5 forward (y+p), 4 back (p), and 7 forward (o) again then your net distance traveled would be 8, but you'd have taken 5+4+7=16 total steps. Total steps is the perimeter. In this case, there is no way to determine the actual value of the purple part and hence no way to determine total steps.
The faster way to solve this is:
1) Assume there is a solution.
2) Then the length of that inset bit must be irrelevant to the solution, since we're not given enough information to determine how large it is.
3) Reduce the depth of the inset until it is arbitrarily small, then the bottom and "top" will each equal 12.
Note that you can also reduce the height of the top bit and the inset to be arbitrarily close to zero, in which case you have a rectangle with sides of 9 and 12. The perimeter is left as an exercise for the student. 8-)
@2:35 “the horizontal lengths will magically work out in a similar fashion” No-there’s nothing “magic” about it. It’s just basic math. When people refer to things like this as “magic”, it causes people with lesser mathematical skills to actually think there’s deception or trickery involved, and to therefore doubt both the legitimacy of the answer and the integrity of the person presenting the answer. We have enough people in the world now whose knee-jerk response to anything they’re personally incapable of comprehending is to declare it “disinformation”, without also having people who are presenting factual truth also undermining their presentation by calling it “magic”.
Basic math is rather magical, though.
I have always been enthralled by the way the numbers magically fit into each other. For instance,
1+4 = 2+3 = 2+2+1 = 5
or
2×30 = 6×10 = 4×15 = 5×12 = 2×2×3×5 = 60.
It is not complicated.
Simple things can be magical (like a sunset, an evening hanging out with friends, playing a game with one's parents, rain falling on a tin roof, etc) without being complicated.
But people who don’t understand math (or “magic”, in the sense of magicians performing show business tricks) equate “magic” with dishonest deception. By calling math “magic”, you are reinforcing that misplaced distrust.
@@verkuilb When someone tells you that an evening was magical, do you assume that they were scammed?
When someone points out that flipping a switch on the wall "magically" turns on all the lights in a room, do you believe that they were deceived?
He said that something magical occurred. He did not claim that math is magic (indescribable). He stated that something wondrous happened.
@@1st2nd2 I don’t-but I’m not talking about me or you. I’m talking about all the billions of simple-minded people in this world, who do think that math and science are literal, deceptive ‘magic’. We need to take affirmative steps to stop reinforcing their delusions.
@@verkuilb Do these hypothetical people believe that when they are told that an evening was magical, there was deception involved?
I don't know anyone who believes that something happening "magically" involves deception. I have worked many years in retail and encountered many people you might label as "simple-minded" (I would not, btw.), and none of them would think that.
Please be careful in labeling people as simple-minded due to believing that people are deceptive. I know plenty of relatively intelligent individuals who have been taken in by certain theories. That does not make them simple-minded.
What does the trick in these kinds of questions is try to change free variables and see if the problem is really impossible, like what happens to the shape and perimeter. In this case, increasing the length of the notch would "pull back" the length of the bottom, since the values of 5 and 7 are fixed, and that immediately tipped me up to the relationship between bottom and notch, which is key to the answer.
Again very nice one, but it could be solved in a much simpler way: once you assume all angles are 90 degree, you’ll realize you can “stretch” the figure to the right compressing the purple segments extractly as much as the red one to the point the purple is gone. Then you realize that you can compress similarly the vertical line green as much as you expand the yellow one. And so on until you realize you are bavk to a simple rectangle of sides 9 and 12 (7+5), which - guess what - has indeed perimeter 42…
Xs dont cancel. You cant solve for X. You are ignoring the middle line of X.
Top Line = (5+7-X)
Middle Line = X
Bottom Line = (5+7-X)
The real problem is 9+9+(5+7-X)+X+(5+7-X) = y [I added in parenthesis to make it easy to read]
42-X+X-X = Y
This is your final answer 42-X=Y
For this guy to get y = 42 then x = 0 which cant be true by looking at the picture. You have to know the value of X as X does not equal 0.
The entire perimeter cannot be solved util you find the value of X which you cant do. You can only get 42-X=Y
@@common_c3nts you can get to the same solution assuming that X=2.5.
The top line is NOT 5+7-X... the top line is just 5. Its labelled.
Middle Line = 2.5
Bottom Line = 5+7-2.5 = 9.5
Thus you get, 9 + 5 + (9) + 2.5 + 7 + 9.5 = 42
where each term is (in order): the first vertical, the top segment, the sum of the right vertical segments, the middle line, & the bottom line. Your mistake is in thinking that the yellow segment on top equals 5, when it actually equals 5-X.
I figured it by imagining extending the base by the length of the incursion and that transform into a rectangle with the same perimeter.
You dope - the purple is already included in the the 7. Your math is off. Try again? You have to subtract the purple to get the horizontal length.
The math is not off. Maybe try rewatching the video. You clearly missed something in Presh's explanation.
@@MuffinsAPlenty you rewatch it - the math is waaaaay off
@@richardleahey8679 Then please explain the correct solution.
This is a very complicated explanation of a relatively simple solution once you get you head around it.
Split the shape into it individual rectangles, note there are two of every individual line so add the numbers together and multiply the result by 2.
I put a big square 9x10 around the perimeter, and noted that the only extra length was the two short horizontal lengths in the wee notch, which had to be 2 long
So the perimeter had to be equal to the hill perimeter square, 2(9x10) +4.
To complete the question you need to give lengths to the sides of the rectangle on the other side of the equation; 6x7.
I love the closed captioning showing "rectal linear" instead of _rectilinear._ 🙂
This is the question the computer in Hitchhikers guide to the galaxy answered.
Left vertical = sum of right verticals. That’s 9+9=18. Now let shortest horizontal =x. Sum of all horizontals is: 5+x+7+(5+7-x)=24. Now perimeter is 18+24=42
Think of the purple section as a sliding section. If the purple section is 0, then the problem is easy as the vertical sides are 9 and the horizontal sections 12 (5 +7).
If you slide the bottom two rectangles to the left, then the purple segment gets a variable length. The bottom orange / yellow line segment, however, will lose the exact length of the purple segment at the same time. So whatever length you have to add for the purple segment you end up subtracting from the bottom segment (the lengths cancel each other out) so the total perimeter remains the same (42).
I thought "you can just slide the thing back and forth", then realized that sliding takes away the same amount that it adds. So i could just imagine it was slid all the way to the left
I initially misunderstood the question as "find the area" and thought it was actually impossible due to insufficient information
Great for working out how much fencing you need, fence posts not so much. Get at least an extra panel (Maybe 2) to allow for the ones you will need to cut for the odd sizes.
The three vertical lines of unknown length adding to 9 is pretty obvious for a lot of people, I think. The way I figured out the other part was to name the sides a through g starting at the top and working counter clockwise. I then quickly saw that c = 5-g+7 so c+g = 5+7=12. Now you have everything you need. Cute puzzle.
This is easily solved by normal algebra. Set the unknown sides to a, b, c, d ,e, f, where "a" is the unknown part of the "5" side (white). Then:
a+ c = 5
a+7 =f
b+d+e=9
Perimeter = 21 +b+c+d+e+f
=42 after substituting in. All the unknowns cancel out.
The three edges on the right-hand side of the shape add up to 9 (the same as the unbroken left-hand side). 9 + 9 = 18. Then, if we consider the overlap between the sides of length 5 and and length 7 as being equal to some unknown value x, that means that the bottom edge of the shape has length (12 - x). So the total of the 4 horizontal edges of the shape is 5 + x + 7 + (12 - x) = 24. Adding the 18 from the vertical edges gives 24 + 18 = 42. And that's the answer! (Presh voice 😄)
I think Presh may have missed the fact that the question's diagram was not just a visual demonstration that the diagram is rectilinear but that it was stating that the perimeter of the first rectilinear shape on LHS is the same as the perimeter of the rectilinear shape on the RHS.
Let p1 be the perimeter of the LHS and p2 be the perimeter of the RHS.
Let X and Y be the lengths of the sides of the rectilinear shape on the RHS.
Then p2 = 2x+2y.
For simplicity, choose one side of the shape (Y) on the RHS to be length 9 such that it matches the LHS.
Then p2 = 2x+18.
Let i, j, and k be the vertical unknown lengths on the LHS in descending order.
Then (i+j+k) = 9.
Let z be the horizontal unknown length between i and j.
Then 5+7-x = z.
Solve for x, x = 12-z.
Then p1 = 9+5+7+(i+j+k)+x+z.
Simplify p1 = (i+j+k)+x+z+21.
By the question p1 = p2.
Then i+j+k+x+z+21 = 2x+18.
Solve for x, x = (i+j+k)+z+3.
Then 12-z = (i+j+k)+z+3.
Solve for z, z = (9-(i+j+k))/2.
Substitute (i+j+k), z = (9 - 9)/2.
Simplify z = 0.
Then x = 12-0.
Simplify x = 12.
Since p1 = (i+j+k) + x + z + 21.
Solve p1 = (9) + (0) + (12) + 21
Simplify p1 = 42.
We arrive the same solution but while Presh's method works without the additional constraint of the question, discarding it leaves potential information on the table. Should a follow-up question have asked about the unknown length we called z he would not be able to answer it without adjusting methodology or acquiring additional information. This is evidenced by the fact that while he could demonstrate i , j, and k could vary without consequence to the perimeter, the same could not be done for z
To be fair to Presh this question was presented poorly by whoever wrote it. It is improper to use an equal sign like shown in the image of the question without sufficient context denoting what is being equivocated or information can be misconstrued. While I am technically inferring that it is the perimeters of the 2 shapes being compared with the equals sign as much as Presh might have. These problems are generally designed to give no more information than what is required for solving it, so we must assume that the addition of the RHS of the equation is not redundant and is adding new, necessary information.
A slight rephrasing of the question would make the problem much less confusing while maintaining intended difficulty. An addition mentioning that the diagrams are not necessarily drawn to scale, while not required, could point some students in the right direction if they are unsure where to start.
All the vertical lines on the right hand side of the figure add up to 9. If the horizontal distance between the two vertical lines on the right hand side of the figure is defined as x, then the perimeter is 5 + 9 + 5 + x + 7 + (7-x) + 9 = 42.
Since we are given that the vertical and horizontal perimeter sections can be easily separated, then the total is then
the sum of the horizontal and vertical sections of the figure can be found separately and then summed. Let P be the total perimeter---
----which is the sum of the vertical and horizontal sections. Pv =9+9 = 18, and Ph=5+7-x where x is the unknown horizontal section. The answer then is: Pv+Ph= 18+5+x+7+5+7-x = 42.
So lets add so notation. The line under 5 and above 7 is x so the last line is 12-x. Then we note vertical lines with a,b,c.
A+b+c= 9
Then we have 2 other unknow lengths of x and 12-x that when added will be 12
So the total is 9*2+12*2=42
I figured the bottom was the two horizontals added together minus their overlap so 5 + 7 - (7 - 5) so the bottom is 10 and the middle unknown is 2 so when combined with the obvious sides you get 9 + 9 + 5 + 7 + 10 + 2 = 42.
Seemed to me to be the easiest way.
1:50 When you have to calculate the perimeter you can do changes, that influence the area but not the perimeter. E.g. like shifting the line marked with 7. Just shift it to the right until it´s left end is under the right end of the line marked with 5. (The next step is to make a rectangle of it, if you need it). So you have 9 + 5 + 7 for half of the perimeter.
I struggled with this one for a bit because we don't know the length of the overlap, and then I hit on an idea - what if the length of the overlap doesn't matter? To test it it, I assumed it was 2 and calculate the perimeter as 42. Then I assumed it was 3, and got the same result. So then I called the overlap x, and set up the equation, and found that x cancels out.
The answer is even easier, if you assume the length of the bottom part is 5+7 then you'll be over counting since 5 and 7 have overlap. so just don't count that part of the perimeter (the purple overhang part) and it will all balance out.
This one was so easy I thought there was some trick to it....there wasn't.
I assumed the bottom side was also length 9, making the shape a square with parts cut out. The perimeter calculation also comes out to 42.
Alternative visualisation is to imagine you walk along the perimeter until you reach your starting point again, let's say in counterclockwise direction. Clearly, we have to walk as far up as we walk down, and also as far left as we walk right. Since we know all edges that we walk along downwards and left, and all edges we traverse upwards or right are unknown, the perimeter is twice the sum of all known edges.
A version of this puzzle was a master puzzle in the game, Professor Layton and the Curious Village.
Just giving it a shot before the vid:
labelled each unknown side length a, b, c, d, e starting from top going clockwise around the shape.
from known sides we get that:
a + c + d = 9
5 + 7 - b = e ----> 5 + 7 = b + e
so combining these we get that
(a + c + d) + (b + e) = (9) + (5 + 7) -----> a + b + c + d + e = 21
so all the unknown sides add to 21, and the known sides also add to 21 (9 + 5 + 7 = 21)
so 21 + 21 = 42 is the answer
hey right answer with slightly different method I'm happy with that
Move the green line in vertical alignment with the blue line, moving the 7 line, and connected lines, with it (so, as the horizontal line between the blue and green reduces, the baseline increases)...
then... move the 7 line, and connected lines, upwards to align with the 5 line, forming a rectangle, 9 high by (5 + 7) wide...
... with perimeter (9 + 5 + 7) x 2 = 42
I was too caught up in finding the values of the missing sides to notice that you only need to find the sum of them. This is a nice way of thinking about perimeters.
This is surprisingly easy... the presentation is just daunting
If you think of it as a belt going around tiny little pulleys, it's easy to visualize stretching this out horizontally while maintaining the perimeter. Then you can pop out that last corner to complete a 9x12 rectangle. Hardly any math required.
how do you 'stretch' it out? Can you explain your process?
Your way was so much simpler than mine. I calculated the two areas (9x7) and (9x5), found them difference which would be the area overlapping (18cm square) and then found it’s perimeter (9x2). So the length of the red side was 2cm. Worked out the horizontal lengths from there and worked out the vertical length as you did. Got the same answer but took way longer. Thanks for the video.
It's cool that you can get the perimeter without knowing the smaller side lengths.
A much clearer approach I think is to label the unknown sides and use substitution.
Let a = the unknown side perpendicular to side of length 5
Let b = the unknown side perpendicular to a
Let c = the unknown side perpendicular to b
Let d = the unknown side perpendicular to side of length 7
Let e = the unknown side comprising the base, i.e. the line segment perpendicular to d and side of length 9
Now the definition of perimeter yields
9 + 5 + 7 + a + b + c + d + e
=> 21 + a + b + c + d + e
We know from the colorful coding that a + c + d = 9 so
21 + 9 + b + e
=> 30 + b + e
e = 5 + 7 - b or, e + b = 12
=> 30 + 12
=> 42
5 is traveled twice and 7 is traveled twice. That overlap made it hard to see. ty
If there is a single answer, which the question asking for a single answer would have you believe, then the wiggle to the left can just as well be zero size as any other size. In which case both top and bottom are 12, and left and right are 9, hence 42.
If you want certainty you can observe that whatever length we increase the wiggle to, an equal amount is added to the wiggle, and subtracted from the bottom.
Looks bad until you start writing down what you know. Perimeter is made up of horizontal components and vertical components. Getting the total vertical is easy, there's no overlap. Pv = 9 + 9
Then write down what you know about the horizontal, with the two unknown segments labeled x (short) and y (bottom). You can express y in terms of x.
y = 5 - x + 7
Then write down what the total horizontal component of the perimeter is.
Ph = y + 7 + x + 5
Substitute in your equation for y, so the horizontal perimeter is only in terms of x.
Ph = 5 - x + 7 + 7 + x + 5
Viola, x cancels out and you're done. Well, add the horizontal and vertical components...