You are in the center of a square room and 25 ft away from a corner. What’s the area of the room?
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- Опубликовано: 7 июн 2024
- How to solve a math word problem topics: squares, area of square, right triangle and Pythagorean Theorem. Learn more math at TCMathAcademy.com/.
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![16 divided by [ (9 - 5) squared ] times 16 =? Can You Solve In Your Head?](http://i.ytimg.com/vi/MBlHXVxSuHA/mqdefault.jpg)
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Draw the diagonal of the large square. The diagonal will be twice the distance from you to the corner, so 50 feet. This divides the square into 2 triangles. Choose either triangle and construct a square on each side of it. The square on the hypotenuse is 50 x 50 = 2500 square feet. By the Pythagorean theorem we know that the sum of the other two squares is also 2500. Further, the sides of the smaller squares will be equal, so their areas are also equal, and will be 2500 / 2 = 1250 square feet. Now notice that the original square had side length exactly the same as the sides of the two smaller squares. Therefore the area of the original square is exactly the same at 1250 square feet.
1250.32
@@sutters7251 What?
You are at the centre of a square, 25' from a corner, so, 25' from each corners. Take any two adjacent corners, and you have a right triangle, with one face of the square as it hypotenuse. So, take 25^2 * 2 to get the square of length of the hypotenuse. So, 25^2 * 2 = 625 * 2 = 1250. You don't need to find the square root of 1250 just to square it. Meaning that you already have the area of the square room : 1250 square feet.
Cool approach. Thanks.
I did it in my head by just knowing that the diagonal of a square is √2 times the length of a side. So 50/√2 is the length of a side. Square that: (50/√2)² = 2500/2 = 1250.
That's exactly how I did it. Seems easier than the method shown in the video.
I did it this way in my head before watching the video - but I build stuff so maybe that helped.
Do not complain if you don't like this teacher. He is helping people that are learning math, not people that already can math.
Keep on teach, teacher 👍
@borjewahlen6917 I totally agree with you and I will add that he is an excellent teacher.
The diagonal of the square room is 50ft corner to corner. Therefore 2 x side^2 = 50^2. side^2 = area = 50^2/2 = 1250 sqft
That's what I did in my head. 🤷♂️
@@Gideon_Judges6 Yup. Not difficult.
It took me less than 20 seconds to read your explanation. I should have gone straight to your comment. Sadly, I watched John’s 18 minute lecture first. It was, how shall we say, an ordeal.
Triangle, area of = base times height divided by 2.
You are 25 feet from the 3 corners of a right triangle therefore the base is 50 feet and the height is 25. You have 2 of those triangles therefore the 2’s cancel, so 25 times 50 is 1250.
A second way is to recognize that a square is comprised of four right triangles that form two smaller squares that have a side length of the distance from the center of the larger square to one of its corners.
25*25= 625 is the area of each small square times 2 = 1250 sq ft.
A) youngster, I enlisted in 1971 and retired in 1995 from the Navy.
B) I fold the flag when we bury one of Uncle Sam’s Misguided Children
(For those of you that are confused, there is branch rivalry, however we band together when an outsider gets involved.)
Easier mental math - from the center to the corner is 25 and also to all corners. So, From the center to two corners forms an isosceles right triangle which is 1/4 of the square. Base times height is 25*25 or 625. The square is twice that value (or (625/2) * 4), or 1250.
@@walkfaster Ok, here's another way. Slide rule enthusiasts should not be left out of the fun.
If you're 25 feet from the corner, the area of the square is 10^(log(25)*2+log(2)) = 1250.
log(25)*2 is the log of 25^2. log(25^2)+log(2) is the log of 25^2 times two.
You raise ten to that power because the base ten log of a number is the exponent you need to use against a base of 10 to get the number.
Note that any log base will do, so e^(ln(25)*2+ln(2)) also equals 1250.
However, log of 25 is hard and any log will do, right?
How about 25^(2+log(2)/log(25))? That also equals 1250, but is trickier.
2 is log base 25 of twenty five times 2, because log base 25 of 25 is 1. 1*2, Terrance Howard notwithstanding, is 2.
log(2)/log(25) is the base 25 log of 2. That converts the base ten log of two to its base 25 counterpart. Addling logs is like multiplying the bases.
Adding it to the log_25(25)*2 is, again, multiplying by 2. You get the base 25 log of 25^2 * 2.
Take 25 to that power and you get 1250.
Now, to my calculator to see if that worked. I'm pretty sure it did.
@@walkfaster You have impeccable taste!
@@johnnyragadoo2414 I had a Pickett, back in the 1960s.
@@darkdelta A EE professor I had was catching grief from a computer major who thought computers would soon obsolete engineering. “Young lady,” he said, “we put men on the moon and returned them safely with slide rules and three digits of precision.” I will never forget that comeback. That was Professor Dougal at UT Austin. An amazing guy.
@@johnnyragadoo2414 Shut herdown pretty ole quick.
I can remember talking with some colleagues, again in the 1960s, how one day we'll get electronic adding machines. We never considered multiplication and division. Then Burroughs came out with a desktop calculator, it sported a Nixie tube display, among other features, Now my phone has more computing power than some mainframes back then,
Two sides of a triangle, equal to each other, with the hypotenuse being third side. Each side is25. So 25 squared plus 25 squared equals the hypotenuse squared, which is the area of the square.
It is a right, equilateral triangle and an hypotenuse of 50. A 1, 1 similar triangle has a hypotenuse of square root(2), or remember this relationship from prior interactions with this triangle. If we chose a similar triangle with a hypotenuse of 50, the sides are the 50 divided by the square root of 2. The square’s area is one side squared or 50 squared divided by the square root of 2 squared or 2500/2. So area is 1250 square units.
@16:40 Keep in mind that X is only approximately equal to 35.35 ft but it is exactly equal to the square root of one half the square of the hypotenuse. The area of a SQUARE is equal to one half of the square of the hypotenuse and will equal the square of one side.
Length of a diagonal of a square= a✓2 (where a is the side of the square)
As such , a✓2 = 50
a = 50/✓2 ft
Area of the square= a^2
= (50 ft/✓2)^2
1250^ft^2/2= 1250 ft^2
My way also.
Diagonals in a square intersect at 90 dgr. So a side of the square can be the hypotenese of a right-angle triangle with equal cathetuses
I looked at it like four triangles.
If you put two of the triangles together you have a square with sides 25 x 25.
25 x 25 = 625 Sq ft
625 x 2 = 1250 Sq ft
I solved it the same way. Took me 10 seconds.
We are so impatient that we tend to forget that we are not the only one in the room. I already know most of what is discussed on this Channel, but I realize that he is catering for the whole class. It's the mark of a good teacher. He makes it easier for the slower one. Let's therefore remember those whose struggle with these concepts.
Hardly anyone realizes it but Pythagoras was one of the Europeans that visited The New World over 2000 years before Columbus. And when he went he took presents for the leaders there. One of the things that most pleased them were hides of animals they could never have imagined. The Chieftains would place these gifts on the ground for their wives to sit on at the ceremonial dinners. One Chieftain was Pythagoras’ favorite and he had given him skins of a Giraffe, a Rhinoceros, and a Hippopotamus. At the ceremonial dinner Pythagoras looked at the Chieftain and his family and a moment of enlightenment washed over him as he saw the wives. Two wives sat on the Giraffe skin, three sat on the Rhinoceros skin and the other five on the Hippo’s skin and he exclaimed, “Eureka! I see it! I understand now! The Squaws of the Hippopotamus is equal to the Sum of the Squaws of the other two Hides!”
Triangle area = ½bh, square = 2 triangles. 2•½•b•h = bh = 50•25 = 10•5•25= 10•125=1250. No calculator required.
A rather easy solution to this problem is if you’re in the center of a room and 25 feet from the corner and the area of a triangle is 1/2 the length times the height then
25×25=625
Now you divide that in half
625÷2 =312.5
And now you have the area of a triangle, that is equivalent to 1/4 the area of the total of the square which means multiply times four and you get the area.
312.5x4=1250
Of course, if you’re thinking ahead of the game then 625 would equal half the area of the room because it’s two of your triangles that would make up the total of the area of four triangles so you could just multiply that by two to get the same answer. Sounds more complicated than it really is. But what I’m basically saying Is calculate the area of a triangle where a side of the square is the hypotenuse of a right triangle where the two sides are 25 feet which is equivalent to 1/4 of the total area of the square.
Except stop after 25x25=625 because you have the area of the triangle that is one half of the square. (Corner 1 to corner 2 to corner 3.) Just multiply that by 2.
@@thomasharding1838 yeah I said that which makes me think you didn’t read everything I wrote.
Exactly how I did it in under a minute. Thanks for the challenge.
The answer is firstly, one has to figure the length of one of the sides of the square. One knows that one is in the middle of the square where one is standing 25 units from a corner. Now, this is actually 50 units from corner to corner and that this length is 45 degrees at a corner. So, one uses the formula cos theta = x/h , base/hypoteneuse. So, x as one of the legs of the square is
x = cos(45 degrees) * 50 and this gives you the leg of the square. Area of the square then becomes ((cos45 degrees)*50)^2.
Form your diagram, Area of a triangle = 1/2 * height (h). Base =50, h = 25, but area of a square = 2* area of triangle. So area = 2. (1/2.50). 25= 1250 sq feet. OR. From your diagram, x= side of square. So, by Pythagorus, x(sq) = 25(sq) + 25(sq). But area of the square = x(sq). So area of square = 2(25(sq)) = 2(625) = 1250 sq feet. I appreciate these 2 solutions are linked as they both give 2(25.25) = 1250 sq feet..
25ft to the centre, so diagonally corner to corner is 50ft. Two sides of the room and that diagonal form a right angle triangle where the other two angles are 45 degrees. The hypotenuse of such a triangle is √2 times the length of each of the other sides (you can check that with Mr Pythagoras) so the side length of the square room is 50/√2.
Therefore the area = (50/√2)² = 2500/2 = 1250 ft²
That's exactly how I did it. This is the second video of John's that I have seen where he did not use the fact that the diagonal of a square is √2 times the length of a side. I think he just wants to show the Pythagorean steps, which is just fine.
The diagonal is 50 but we want a side length to find the square's area, so draw another square around this one (in your head of course) turned 45 degrees so that the corners of our square are in the middle of the sides of the new one. We have the side length for the new square as a given (near enough) 50'. So the area of that square is 2500 sq ft, and it is twice the size of our original square so our square is 1250 sq ft.
Takes much longer to write than it does to work out.
Easy mental arithmetic.
1/2 x base x height = area of a triangle. 1/2 x 50 x 25 = 625. We have two such triangles. 625 + 625 = 1,250 ft^2.
Basically boils down to the same idea behind the question to double the size of a square by cutting it into 4 triangles and mirroring all triangles on the squares' edges.
Thank u
So many ways to solve but depending on the setting e.g in an examination setting speed should be considered!
Very easy.the double of 25ft will be the diagonal of square. So diagonal=50ft
Now the formula of diagonal of square is s root 2 where s is the side of square. We alr know diagonal so from this formula we can find side
S=50/root 2 we know formula of area of square is s×s so the area is equal to 50/root 2×50/root 2 =2500/2=1250ft. Thanks
So many ways to solve this. I used cosine 45 = 0.707 X 25 which is 17.675 for half of one side. Double it to give 35.35 for one side. Square that to give area of 1250ft^2
d/2 = 25 ft, area = d^2/2 square ft
= 625 /2 square feet
Area if a parallelogram is 1/2 * the product of the diagonals. 1250 sq ft.
Yes, I can!
the problem is much simpler that you made it... apply unit circle principals to obvious: you have a triangle with 45 degree angles because opposite and adjacent sides are the same. What does that mean.... x and y scalars to the hypotenuse is sqrt(2)/2 ... and the sqrt term gets squared... you don't have decimals to worry here and the arithmetic is simple.
Did this in my head in about one second. If it's 25 feet to a corner, then a diagonal is 50 feet. So the room can be looked at as a diamond inside a 50 foot square. A diamond has half the area of a circumscribed square. So half of 50 squared is 1250.
It's easy to forget that this problem doesn't ask for the length of the sides. Rather it asks for the area of the square. So once we have a side as sqrt of 1250 we can simply square that for our result. 1250 Sq. Ft.
Start using the metric system, please. :)
Solved at the thumbnail, the room is 35 ft 4.26 in square.
You can work this two slightly different ways usning the Pythagorean Theorem.
1) Given that the room is square, you could take do a right triangle of one quadrant with two 25 ft sides.
25^2 = 625, 625 + 625 = 1250, sqrt(1250) = 35.355.... or 35 ft 4.26 in.
2) The other way would be to do a right triangle of half the room. Double the 25 ft distance to the corner to get the hypoteneuse of 50 ft; 50^2 = 2500. 2500 / 2 = 1250. sqrt(1250) = 35.355....
...and I didn't read the problem 3 times! Should have stopped at 1250, 35.355... squared.
Whoops!
this is easier . 2 diagonals makes 4 r/a isosceles triangles each side 25 ft.. take 2 opposite triangles and put them hypotenuse to hypotenuse and you have a square of 25 ft. side / same with other 2 triangles. 25 squared equals 625 times 2 equals 1250sq.
25ft at the centre of a square so you have the base and height of 4 triangles that make the total area. 25*25/2 then multiply by 4 triangles. Simplifies to 25*25*2. You actually get the correct answer here rather than having to round.
Try X=35.35534 on an 8-digit calculator. It will return exactly 1,250. If you try it on a bigger calculator, you will be over on the 9th digit. (1,250.00006651 on a 12-digit calculator)
You can also sine and cos 45 and hypotenuse as 25 ft
I mean 50
You can also use pythagorean theorem and a=b and c =50 to get one side
Each side S of the room is clearly ( [Root-2] / 2) [ 50 ] ft, so that Area = S^2 = 1250 ft^2
Let x be the side. 2x^2=2500 by Pythagoras theorem
x^2=1250
Area =1250
1250 sq ft 4 orthogonsl triangles 25 ft per side
50^2 / 2
a square has 90° corners
a square has a corner to center line with 45° angle at corner
therefore, the triangle from corner to center to mid-side back to starting corner is an isosolese having angles 45, 45, 90. The sides are of proportion:
1:1:sqrt(2).
In this problem, 25ft relates to sqrt(2). The other two sides are, therefore:
25/sqrt(2) = 25(sqrt(2))/2
=12.5sqrt(2)
This length represents half of the squares side, so that the side's length = 25(sqrt(2))
Area = side×side
= (25(sqrt(2)))^2
= 625×2
= 1250ft^2
verify❌️
corner to center
=sqrt((25/2)^2+(25/2)^2)❌️
=sqrt(156.25+156.25)❌️
=sqrt(312.5)❌️
=17.68 ❌️=25
Verify pt2:
25sqrt(2)/2)×(sqrt(2)
=❤25✔️
review:
for an isosolese triangle
a^2+a^2=h^2
here h = 25
a^2+a^2=25^2
2a^2=625
a^2= 312.5
a =17.68
for bigger square
s=2a
=2(17.68)
=35.36
area = 35.36^2
= 1250ft^2
You did some extra work, though. Since you knew about √2, you could just start by knowing that the diagonal (50) is √2 times the length of a side of the square.
Side of square is 50/√2. Square that for 2500/2 = 1250.
got it 1250 easy 2 isos triangles base = 50 & hight = 25 1/2(bh) 625 x 2 = 1250 or just the initial 1250 as it is 2 triangles thanks for the fun
You sure made it hard. Two triangles at 1/2 bh.
1250 ft^2
If each side is the same in a square. How did you get a = 3, and b = 4; if they are the same in a square?
Didn't say the sides were 3 and 4. He was using the example of a "3,4,5" triangle to demonstrate the Pythagorean theorem with an easy calculation. He then what on to say that I'm the case of the square BOTH "a" and "b" could be replaced by "x" since the sides of a square are all equal (as you said). Why he decided he had to demonstrate the a^2 + b^2 = c^2 relationship rather than assuming the theorem would be self-evident, I don't know. For anyone not familiar with the theorem it was probably more confusing to introduce the 3, 4, 5 example than to just state the theorem and move on (i.e., he likely confused the very people he thought he was helping). You probably weren't the only one who missed the nuance.
There is no need to find the value of x unless you are trying to verify the answer!
Surely 25^2 x 2 is much easier?
25x25:2x4=1250
solved this mentally in about two minutes.
1248 sq. units
L times w times height
1250 sq.ft.
1,225
So why couldnt yo do that using the 25ft and putting the 90° in the centre?
I got 1249.6225 sq. with 35.35 on the side.
Greetings. Yes we can. The answer is 1250 square feet. We have several trigonometric methods at our disposal that can be utilized to determine the answer. Let us assume that the area of the room is in the form of a square ABCD and that there is a diagonal running from A to C. Based on the details given, AC, the diagonal is 50 ft. Additionally, notice that the diagonal disects angle A, 90 degrees, into two equal parts, 45 and 45 degrees. Now, if we further assume that BC is equal to X ft, then Sin 45 = X/50 ft, and X=50 Sin 45=50(.70710678)=
35.35533906 ft. Moving forward, the area of a square is the side squared. Therefore, to find the area of the room, we will simply square the side to get (35.35533906)^2 to get 1,250 square feet.
Alternatively, recognizing that triangle ABC is an isosceles triangle, AB equals BC, and angle ABC equals 90 degrees . If we assume that AB is X ft, we could have said that
X^2+X^2=50^2, 2X^2=2,500 sq. ft.
That is X^2=2,500/2 sq. ft, and X^2, the length of one side is 1,250 sq. ft.
116.1 square metres
1200sq feet
1250 sf
2x1/2x50x25=1250
What about 25x25=625. 625÷2=312.5. 312.5x4=2500 so much simpler..... And it comes up exactly right... Not almost right like your equation
Just use trigonometry. It’s easier.
1250 sqft
Easier 50x25 =1250 Area
Area = 50 ft square
Split the square into two isosceles triangles with a base of 50 and a height of 25.
Each triangle has an area of half base times height. 25×25=625
Thus the area of the square is 2×625=1250.
Forget Pythagoras. Visualize the problem first then simplify it.
No paper or pencil needed. Save the planet.
YES! This Works! Forget Pythagoras not needed this is a fifth grade problem over complicated explanation for no true purpose
1250 sq ft
1250
625 Sq.feet
625
1250 square feet
1250 sq. Ft
easy...solve the side using pythagorean theore.m....then you get the area....elementary, my dear watson...
2500 square feet
100 sq ft
Only a pedant would go about solving this problem with the drudgery you present.
5000 sq ft.
its wrong the correct is 1250 sqft.
Sorry
I think it would have been better not to calculate the approximate square root of 1,250. You want to teach people to think logically. If you have to square the square root of a number, the answer should be obvious.
Your solution is less confusing than this teacher's solution.
The solution is very confusing. I got lost halfway into it.
It is simple. Let the side be X feet. You r from the corner 25 feet away. Which means the hpotenuse is 25feet. The perpendicular let that be X so that your side is 2 C. So hypotenuse square is Xsquare plus the side which is 2x. It is 2X the whole square plus the perpendicular which is X square. So 5 Xsquare. So the area is 25×25. =625 sq feet.
Pythagore theorem😂
18 minutes to remind people that the diagonal of a square is √2 times the side length ... you're a teacher ?
You are making a big confusion
25 FT?
at least in science use the metric!!!!!
54th like
Instead of the 15 minute version....this should have been done under 2 minutes and not 17 minutes.
About 20 seconds for the square and the diagonal that would be 25 feet from the center, and having a length of 50 feet. Another 20 seconds for the right triangles, and the Pythagorean theorem. Another 20 seconds for noting that asquared plus bsquared =c2, which =2500. Another 20 seconds for noting that in a square the sides are the same length, so therefore 2xsquared =2500 and xsquared =1250. Another 20 seconds for taking the area as being xsquared, which is 1250. That is not 17 minutes, and teaching inefficiently is a terrible disservice.
Not sure what viewers are taking on the teacher… if you know more than the teacher move on cuz I’m not interested in your brilliance. Start your own You Tube channel!!
would be nice if you would learn to stop rambling while teaching
1250 sq ft
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1250 sq ft