mathschallenge.net

Problem

What are the dimensions of the shaded square inside the 3-4-5 right angle triangle?

Solution

Let length of the side of the square be, a.

By similar triangles, ^(3a)/_a = ^a/_(4a)

(3a)(4a) = a²

a² 7a + 12 = a²

7a = 12 a = 12/7

Find the position of the square that has the maximum area.
Can you generalise for an x-y-z right angle triangle?
What would be the dimensions of the square in terms of x, y, and z?