Two vertices of a square are located on the hypotenuse of an isosceles right triangle, and the other two vertices are on the legs.

Find the side of the square if the hypotenuse is equal to $a$.
Let vertices $L$ and $M$ of square $K L M N$ lie on the hypotenuse $A B$ of right triangle $A B C$, and vertices $K$ and $N$ lie on the legs $B C$ and $A C$, respectively. Then $B L=K L=M L$ and $A M=M N=M L$.

## Solution

Let vertices $L$ and $M$ of square $K L M N$ lie on the hypotenuse $A B$ of right triangle $A B C$, and vertices $K$ and $N$ lie on the legs $B C$ and $A C$, respectively. Then $\triangle A M N$ and $\triangle B L K$ are also isosceles right triangles, so $B L=K L=M L, A M=M N=M L$. Therefore, $M L=1 / 3 A B$.

## Answer

$a / 3$

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