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Advanced Math/Simple AMC question


The website doesn't have the answers to this question. Could you confirm whether I'm right or wrong, and if I'm wrong explain why and how to do the problem?

I think the answer is e. because x can change the length values and, because there is no set lengths, the triangle can conform.

Line segments drawn from the vertex opposite the hypotenuse of a right triangle to the points trisecting the hypotenuse have lengths sin x and cos x, where x is a real number such that 0<x<pi/2. The length of the hypotenuse is:
a. 4/3 b. 3/2 c. (3(5)^(1/2))/2 d. (2(5)^(1/2))/3 e. Undeterminable

"because x can change the length values and, because there is no set lengths, the triangle can conform."

There are two values in question. There is no reason to believe that being able to change x will simultaneously make sin(x) be one value and cos(x) be the other.

Refer to the attached image.

The hypotenuse is split into c/3, c/3, and c/3 by these lines (red) with length sin(x) and cos(x). However, one can draw perpendicular lines to the legs from those points to create two triangles (blue) similar to the original.

They have side lengths a/3, b/3, and c/3. That means the *other* right triangles have side lengths:

2a/3, b/3, sin(x)


a/3, 2b/3, cos(x)

And if you use the Pythagorean theorem on those, you get:

(4/9)a + (1/9)b = sin(x)

(1/9)a + (4/9)b = cos(x)

Add these two equations to get:

(5/9)a + (5/9)b = sin(x) + cos(x)

Now, the right hand side is 1 by Euler's identity. The left hand side is (5/9)c since this is a right triangle originally. That gives:

(5/9)c = 1

And so c = 3/√(5) or 3√(5)/5. You should check because it looks like maybe you wrote down answer choice (c) incorrectly.

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Clyde Oliver


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