A small triangle is inside of a large triangle - what’s the missing side of the small triangle?
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- Опубликовано: 2 окт 2024
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Easiest way: There is a small triangle in the top right corner, 4 long, 3 high (9 - 6) and with a hypotenusa of 5. The similar triangle on the left is 6 high, which is twice the hight of the small triangle -so since the hight and length are proportional in all these triangles- the length (X) of the left hand triangle must be twice the length of the small triangle, being 4 x 2 = 8. Done.
I surrender to you!
On a hunch I tested for 3-4-5. X2
6-8-10
X may be 8.
Big triangle is 8+4 = 12
Big triangle may be 9-12-15 which is 3-4-5 x3
X=8 is thus proved.
Your way is better.
That was my first thought as well. I guess I’m not as dumb as my grades would have shown😉.
@@richardhole8429 Thanks!
I came here to add the same comment. 3-4-5 triangle on top means 6-8-10 triangle on left. Good job!
@@tomspallone1868 nobody is dumb who keeps on learning
Just using ratios, from 9 to 6 in height is 2/3 then the base 4 is 1/3. Therefore x= 2x4 or 8
TYPO!! The ratio of 6 to 9 is 2/3. Boy, I sure hope I never make one. But you are exactly right.
9-6 = 3. (Take Off Astronaut -> TanO = Opp / Adj) TanO = 3 / 4 = 0.75. -> (Adj = Opp / TanO) x = (Opp / TanO) - 4 x = (9 / 0.75) - 4 -> x = 12 - 4 -> C) 8
You are using ratios -> 3/4 -> Tan = 0.75. Which is basically what I did but using trig. Give me a triangle and I'll always go straight to trig (or the Pythagorean theorem)
Using the proportional approach gives 9/X + 4 = 6/X => 9X = 6(X + 4) = 9X = 6X + 24 => X = 8.
the magic is:
they are similar triangles:
google:
Two triangles are similar if they have the same shape but not necessarily the same size. The corresponding angles are equal, and the corresponding sides are proportional. We can think of one similar triangle as an enlargement or a reduction of the other.
I used, out of desperation, Tan Theta = Opp/Adj. This gave two related equations for Theta, then I solved for x. x =8.
Saw the 6, saw the right angle. That makes the little triangle a 3,4,5 triangle, with the 3 doubled. Therefore x is the 4 doubled. Answer is therefore x=8 and the large triangle is irrelevant.
A right triangle can have a height of 6 with any number of sizes for the base and the hypotenuse. The base could be 4 and the hypotenuse 7.2111. Without the larger similar triangle or the values of the acute angles, you cannot establish the ratio of the sizes.
Is it wrong to use trigonometry? The acute angle theta of both triangles is the same, so you can use the equation tan (theta) = opposite/adjacent and get the same equivalence statement 9/(x+4)=6/x to get this solution x=8
Not wrong, but long winded. Just use proportions.
Pytaghorean number 3-4-5
The height is 9 which is 3 x 3
So the base is 4 x 3 = 12
The x = 12 -4 = 8
I saw it as a 3 4 5 triangle. Used that to come up with the answer.
Which 3-4-5? . . . I drew a horiz line near the top of the triangle which horiz line has same length (4) as the || horiz line at bottom. Then vert. leg of new triangle is 3 (9-6). Hypot of that triangle is 5 (3-4-5). Using the 3 and 4 leg of new 3-4-5 triangle, my ratio then is 3/4 as 9/x+4 or 3/4 = 9/x+4. Cross multiply and you get 3x+12=36; x=8.
Me too bruh i see 3 4 5 triangle eith x3 size then i know the side x+4 = 12
so x = 8 and the answer is c
Funny how we all resolved it differently although based on the same geometrical way of thinking.
I identified 3 triangles,
having the sides 3 and 4
As a = 9 - 6 and b = 4,
leaving a’ = 6/2 and b’ = 4,
thus x = 2b’ or 8
It is not a solution for a math problem , is a lesson of how to go around and around repeating the same concept over and over
But small and big triangles has the same angles. This means
Tan @ of small = Tan@ of big
6/X = 9/(4 + X)
6 = 9X/(4+X)
6 ( 4 + X ) = 9X
24 + 6X = 9X
3X = 24
X = 8
the way I approached this. Was to crate another triangle at the top with the base 4 and the perpendicular height (9-6)=3.
So the triangles height to base ratio of 3:4, giving 3/4= 9/12=9/(x+4) , 12=x-4, hence x=8
Going from right to left, every 4 on the x you lose 3 on the y. So, move 8 more and you lose the last 6 and intersect zero. So 8.
Using the Pythagorean Thereom the smaller triangle is a 6,8,10 unit triangle with that the larger triangle is 9,12,15 making the other missing piece 5 units
You should have seen my face when the light lit up! A problem that seemed impossible suddenly became so easy it seemed basic.
I created a third similar right triangle with a horizontal line off the intersection of the 6 vertical and hypotenuse. We know the adjacent is 4 and the opposite is 3 (9-6) so this is a 3,4,5 triangle
Since the smaller original triangle is 'similar' it too must be a multiple of a 3,4,5; and since the opposite side is 6, this is a 2 times multiple of 3; therefor x is also a 2 times multiple of 4 which means x=8.
The value of X is pretty evident from the first glance and we need just one of the triangles to solve the problem.
Copycat! You just typed over my solution from an hour earlier :-)
Brsvo
@wRAAh Lol! Bet more than a few of us came up with a version of this. Actually writing a coherent explanation took much longer.
@@idamcneill8005 Hehehe I know 🙂
As it is about ratios, I noticed that the vertical sides are in the ratio of two thirds to three thirds, making X equal two thirds of the base, meaning 4 must equal one third, so x equals 2 times 4, equals 8 - much quicker than math man's drawn out explanation! Time is money (or of the essence).
I got it straightaway using the 3, 4, 5 principal.The larger triangle is 12,9 and 15 (the hypotenuse).
Hey I liked this one. True I struggled in Geometry but I was A student in algebra. I forgotten about similar right triangles
Similarity works, but why couldn't you get the answer from (x +4) squared + 9 squared = Hypotenuse squared. x is going to be square root hypotenuse - 4
I did it a different way. x/(x+4) = 6/9 → x/(x+4) = 2/3 → Cross multiply so that 3x = 2x+8 → x = 8 QED
Critics he is using algebra to solve this, not geometry, ratio and proportion. That's what he is teaching here
Yeah, I thought that was interesting that there are multiple ways to solve it, and I didn't think of his way. But you still kinda have to know about the geometric proportions in order for the algebra to work. It's all interconnected, which is the fun of math.
This guy talks too much!! He had said the same thing about a dozen times!! I would hate being in his class!
Thank you
Could he have explained it in half the time?
This is why high school students fall asleep
6 is 2/3rds of 9, so x is 2/3rds of (×+4), which means 4 is 1/3 of (× + 4) so x as to be 2 x 4 = 8. The numbers given were small enough to do in my head, could visualize this in my head, but the key word is proportional. 6/9 = 2/3, 4 is 1/3 of (x + 4) so x has to be 2/3. 2 × 4=8.
Obviously, bigger numbers would be more complicated and I'd need paper!
Complete the rectangle in the triangle. This makes a smaller triangle at the top with a base of 4 and height of 3. Since all triangles in diagram are similar, you have a 3,4,5 triangle, a 6,8,10 triangle and a 9,12,15 triangle. X=8!
They'e similar, because their lines are all parallel and they have one common angle. So the ratios will be consistent. So 6 is to 9, as X is to X + 4. Since 9 is 1.5 of 6, X +4 must be 1.5 of X, so X must be 8. I don't think we know it's a 3-4-5 because we don't have enough info. Eyeballing it doesn't count.
By definition of Right triangle
a^2 +b^2 = c^2
3 × 2 = 6
4 × 2 = 8
5 × 2 = 10
Larger triangle
3 × 3 = 9
4 × 3 = 12
5 × 3 = 15
We have all the info we need by the definitions of Right & Similar triangles
He talks to much and is so boring
Feel free to skip forward
John, Just want to say that lately your audio seems to be very low. Thanks for another interesting problem :)
It is truly amazing that you have such a great passion for math AND teaching it ! What a wonderful person. Thank you !!!😊
Wow!! You made that difficult. Since this is multiple choice and more than likely an exam question it is ok to do a quick substitution. The increase in size of the sides is 6/9=2/3. The obvious choice is 8 as the mathematics are easiest and it looks about right. The increase in base length is 8/(8+4) = 8/12 = 2/3. As a final check 6/8 = 0.75 and 9/12 = 0.75. In exams, students waste too much time doing the long method you just did. As a student I would have liked to be told that substitution is a valid method. Cheers
You can solve it that way. Or, if you actually want to learn maths, you can study the correct method.
I understand how to do it the formal way. But if you want to answer as many questions as you can in an exam you have to be a bit smarter. I still used similar triangles but did it a different way. There is no right or wrong way as long as you arrive at the correct solution.
The corresponding sides of similar triangles have the same ratios. Therefore x/(x+4) = 6/9. You can reduce the 6/9 to 2/3. x/(x+4) = 2/3. When you cross multiply, you end up with 2(x+4) = 3x. or 2x + 8 = 3x. Subtract 2x from each side and you end up with x=8
The answer is c) x=8, because x/6=(x+4)/9
In german,it is called "Strahlensatz", what i have used tto get the answer.
It is not necessary tto know, tthat the triangels have a right angel, tthhe onl thhing, ou have tto know, is tthatt the lines witth lengths 6 and 9 are parallel.
You should be able to answer this just by looking at the problem and solve it in your head
The first time I ever solved problems like this was using similar Triangles This was long before I took Algebra 2 or Trigonometry Having taken Trigonometry Yes there are other ways to do this problem
Mr john you say a side is missing in a triangle, then how can it be a triangle
Yap yap yap. get on with it!
Greetings. The answer is 8 units. Using the properties of similar triangles gives 6/9=X/(X+4), cross multiplying gives 9X=6X+24, and
9X-6X=24 after collecting like terms. Moving forward, 3X=24, dividing both sides by 3 gives X=8. Therefore, the answer is 8 units.
From the thumbnail and without watching (using proportion of similar triangles): 12/9=8/6; 8+4=12. The missing side is 8. The only real question is: why is the video 15 minutes long?!?
Seeing its been about 30+ years since I was in a geometry class and don't remember much, I figured to find the slope of the hypotenuse (9-6)/4 = 3/4 so for every 1 unit on the vertical the horizontal will change 4/3, so 6*(4/3) = 8. The fact I was able to think this trough to the correct answer that was an off the wall way to get there makes me wonder why I could never do this when I actually needed to be able to in school. If I had this in school and didn't remember the similar triangle thing I would of just guessed and moved on.
HOLD ON: Now in this video when you got t6o the Sqrt(25) you said the answer was +/- 5 and even wrote it on the table. In all of the other videos, you lecture about the Principal Square Root and say that the +/- X is wrong.
8. 3 4 5
Anybody who knows their triangles should be able to do this one in their head in about fifteen seconds. No doubt others have said so already but i am not going to read every comment. This is a 3,4,5 triangle you can see that if you imagine a small triangle at the top of the big triangle extending a horizontal line from the top of the small triangle, 9-6=3 for its perpendicular, the base is given at 4 looking at the large triangle the base is 9/3*4 =12
x=12-4=8
The ratio of corresponding sides of similar triangles will be the same for all side. The ratio of the 6 to the 9 side is the same as the X side to the X + 4 side. 9 = 1.5 times 6 so X + 4 = 1.5 times x. => 1/2 x = 4 => X = 8
1/- the ray of the sun
2/- a bird on the right-top
3/- the number of the windows under the red roof of the castle
4/- the railing on the top of the room on the right
5/- pattern above the window of a room on the left
6/- left ear
7/- the shape of the yellow flower on the mane
8/- the mane on the right shoulder
9/- tail
10/- a yellow flower on the right grass
6 is 3 less than 9. 3 more than 9 is 6*2. It would then be the same shape, flipped over, and turned the opposite direction. 6•2=X+4.
C. Since they are similar triangles and the 6->9 that means that the 4 is the same proportion to x as 9-6 is to 6. a'la' 8.
I think if you tried you could have taken twice as long to formulate an answer,,,
Lol. I just did it in my head. Notice that they are 3,4,5 triangles. 4X2 =8. EASY.
Same here, by making a square producing a mini 3,4,5 at the top, the perpendicular height of the bigger triangle is 6 which is double the small giving a 6, 8, 10, .
I am astonished how long you can draw out your explanation in order to make a 30- second question take 15 minutes to explain...
So ..
X is to 6 as X + 4 is to 9
6/X = 9/(X + 4) cross multiply
6X + 24 = 9X
24 = 9X - 6X
24 = 3X
8 = X answer c)
It has to be 8. As far as I remember, both triangles have the same ratio, oops, proportion.
Ratios seem to be a simple way to tackle this. 9 is to 6 as x+4 is to x.
Similar Triangles YOU
CAN solve this using ratio and proportion I learned to do this in Geometry Long before I Took Trigonometry
I did this type problem in Geometry We solved it by similar triangles In order to take Geometry you had to pass Algebra 1 Similar Yeh we definitely used Algebra in Geometry
I got the answer 8. Total Area=1/2(x+4)*9, A1=1/2(4*3)=6,A2=6*4=24,A3=6*X Therefore, Total area=A1+A2+A3. Thus it will be calculated as follows: 1/2(x+4)*9=6+24+1/2(X*6), after some steps,4.5X+18=30+3X. The value of x will be 8
x/(x+4)=6/9,x/(x+4)=2/3,3x=2x+8,x=8 so Answer (c)
Yay! Got it right with some mental numeracy….rare for me😂. All about proportions and an assumption of being a 3/4/5 triangle. Which since it is multi choice is ok..or at least a pass!
8
I spotted the 3 4 5 triangle in just a few seconds. So the left triangle must be a 6 8 10. So the answer is 8.
Suppose it was one not easily spotted. Does the equation not become handy?
8
8
At the title (0:01), my answer is c) 8.
6 and 9 have a 2:3 ratio which should carry through the base as well, and 3 * 4 = 12.
X8×6÷2=24 x aria
x / x+4 = 6/ 9 by proportion
Solving x = 8
Do not need ratios. The large triangle is a 9,12,15 triangle. 12 - 4 = 8.
But how do know it’s 9,12,15?
Bat solution gigen
X=8
You take too long to describe it you lost me
X=8
It’s so easy to solve
3, 4, 5 triangle
volume is low.
Einfach den Strahlensatz anwenden.
C
Easy 8
got it 8 simple proporition thanks for the fun.
None of the triangles have missing sides.
😮9/9=4+x/x
X=8
Y•3/2=X•2/3?
8.
8
8
ask MM the time he will build you a watch
c.)8
❤
8