Math Olympiad | Find the area of the square inside the triangle | 2 Different Methods

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Math Booster

Math Booster

Ай бұрын

Пікірлер: 12
@SALogics
@SALogics Ай бұрын
Very nine explanation
@stevenjeng1237
@stevenjeng1237 Ай бұрын
It is easier to let AP=x, then QC=10-2x=5x, so x=10/7, closed
@hongningsuen1348
@hongningsuen1348 Ай бұрын
Alternative method using trigonometric ratio: 1. BC = sqrt 125 (by Pythagoras theorem) 2. Angle ABC = angle APQ (by corresponding angles of // lines BC & PQ, or by similar triangles ABC ~ APQ) Let the angle be theta. 3. From triangle ABC, cos(theta) = 1/sqrt 5 and sin(theta) = 2/sqrt 5. 4. Let side of square PQRS be x. 5. In triangle PBS, PB = x/sin(theta) = (x sqrt 5)/2 6. In triangle APQ, AP = xcos(theta) = x/sqrt 5 7. PB + AP = 5 (given) Hence (x sqrt 5)/2 + x/sqrt 5 = 5 x = (10 sqrt 5)/7 8. Area of square = x^2 = 500/49.
@lasalleman6792
@lasalleman6792 Ай бұрын
Or side of square * (10/5 +5/10 +1) = hypotenuse. Here 11.18/3.5 ; side of square: 3.19 Area of square 10.20
@prime423
@prime423 Ай бұрын
The most important part is not the mechanics but the plan., Here we have 4 similar triangles plus the fact PQ,a side of the square is parallel to the base of the large triangle. Can we use these observations to solve this problem? The answer is yes with more than one solution.
@murdock5537
@murdock5537 21 күн бұрын
(10√5/7)^2
@jimlocke9320
@jimlocke9320 Ай бұрын
We find that ΔABC, ΔABC, ΔAPQ, ΔPBS and ΔCQR are similar. Lengths AB and AC are given and length BC is found by the Pythagorean theorem to be 5√5. The ratio of sides for the similar triangles (short : long : hypotenuse) is 5:10:5√5, which simplifies to 1:2:√5. As in the video, let the length of the side of the square be x. Then, by ratios of sides of similar triangles, AP = x/(√5) and PB = x(√5)/2. However, AB = 5 (given) and AP + PB = AB. So, x/(√5) + x(√5)/2 = 5. Multiply both sides by √5 and simplify: x + 5x/2 = 5√5, 7x/2 = 5√5, x = (10√5)/7. Area of square = x² = ((10√5)/7)² = 500/49, as Math Booster also found.
@michaeldoerr5810
@michaeldoerr5810 Ай бұрын
This is the fifth video in which I have learned both methods and have found both methods more useful than the comments. I shall put that problem as geometry practice.
@quigonkenny
@quigonkenny Ай бұрын
Edit: Watching the video after my post, at 14:25, you can save a step when comparing ∆ABC and ∆APQ. You don't need to use similar triangles to find a value for AM with respect to x. You already have one, and with it you can directly calculate x: BC/PQ = AN/AM 5√5/x = 2√5/[2√5-x]
@devondevon4366
@devondevon4366 Ай бұрын
10.209 approximately 10.209 The presence of the square turns all 4 triangles into a similar triangle Hence, all the triangles are similar Notice that one leg is twice the other (5 vs. 10), or one is half the other. Using Pythagorean, the hypotenuse = 11.10834 Let label the square n. label all four sides Hence, the smallest leg of the triangle to the left = 0.5 n and the long leg other triangle to the right = 2n Hence, the hypotenuse of the large triangle in terms of n = 0.5n + 2n + n = 3.5 n But 3.5n = 11.1834, the length of the hypotenuse Hence n= 11.1834/3.5 n= 3.193811 Hence, the area of the square = n^2 or 3.193811^2 = 10.209Answerr
@devondevon4366
@devondevon4366 Ай бұрын
16:32 500/49 = 10.2
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