## Maximum Square

#### 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)}

(3`a`)(4`a`) = `a`^{2}

`a`^{2} 7`a` + 12 = `a`^{2}

7`a` = 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`?

Problem ID: 152 (Feb 2004) Difficulty: 2 Star