[ Rectangles and Squares. Properties and Characteristics ]

A rectangle is inscribed in a square such that one vertex of the rectangle lies on each side of the square, and the sides of the rectangle are parallel to the diagonals of the square. Find the sides of this rectangle, given that one of them is twice as large as the other and that the diagonal of the square is 12.
Prove that the diagonal of a square is equal to the semi-perimeter of a rectangle.

## Solution

Let the vertices $M, N, K, L$ of the rectangle $M N K L$ be located on the sides $A B, B C, C D, A D$ of the square $A B C D$ respectively; $M N=2 K L$; the side $M N$ is parallel to the diagonal $A C$ of the square; $P$ and $Q$ are the points of intersection of $A C$ with the opposite sides $M L$ and $N K$ of the rectangle $M N K L$. Let $M L=N K=2 x, M N=K L=4 x$. Then $M P=N Q=x$.

Since $A P M$ and $C Q N$ are isosceles right triangles, we have $A P=M P=x$ and $C Q=N Q=x$. Therefore,

$$
P M+M N+N Q=A P+P Q+Q C=12 \text{, or } x+4 x+x=12 \text{. }
$$

$$
M N=4 x=8, K N=2 x=4
$$

## Answer

4 and 8.