Right triangle ABC has side lenghts AB = 3, BC = 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.
+876 $\triangle ZYC \sim \triangle WBZ \sim \triangle ABC$
Let the length of the square be $s.$
$\frac{CZ}{s} = \frac{5}{3} \Rightarrow CZ = \frac{5}{3} s$
$\frac{ZB}{s} = \frac{4}{5} \Rightarrow ZB = \frac{4}{5}s$
$\frac{5}{3} s + \frac{4}{5}s = CZ + ZB = 4.$
$25s +12s = 60$
$37s = 60$
$s=\frac{60}{37}$
