Right triangle ABC has side lenghts AB = 3, BC = 4, and AC = 5. Square XYZW is inscribed in triangle ABC with X and Y on line AB, and Z on

line BC. What is the side length of the square

xMysticalLegend Nov 21, 2020

#1**+4 **

This is an interesting answer:

Using similar triangles twice, and calling the length of the side of the square x,

in the triangle CYZ, CY/x = 4/3, so CY = 4x/3,

in the triangle WXA, x/XA = 3/4, so XA = 3x/4.

The hypotenuse of the triangle is

CY + YX + XA = 4x/3 + x + 3x/4 = 37x/12 = 5,

so x = 60/37.

(Solved by Guest - Nov 13, 2015)

jugoslav Nov 21, 2020

#2**+4 **

Call the side of the square = s

Triangle ABC is similar to triangle AXW

So

AB/BC = AX /XW

3/ 4 = AX / s

AX = (3/4)s

Triangle ABC is also similar to triangle ZYC

So

AB/ BC = ZY/YC

3/4 = s/ YC

YC = (4/3)s

And

AX + XY + YC = 5

(3/4)s + s + (4/3)s = 5

(9/12)s + (12/12) + (16/12)s = 5

(37/12)s = 5

s = 5(12/37) = 60/37

CPhill Nov 21, 2020