Question

A square inscribed in a right triangle with legs $a$ and $b$ has a common right angle with the triangle. Find the perimeter of the square.

Answer 1

Let the side of the square be $x$. The legs of the right angled triangle be $a,b$ and the hypotenuse be $c$.

The area of the triangle $=\frac{1}{2}ab=\frac{1}{2}x(a-x)+\frac{1}{2}x(b-x)+x^2$

$⟹ab=x(a+b)$

$⟹x=\frac{ab}{a+b}$

Thus the perimeter of the square $=4x=\frac{4ab}{a+b}$