[ Rectangles and squares. Properties and characteristics ] [ Criteria for the equality of right triangles ] $A B C D$ is a rectangle, $M$ is the midpoint of side $B C$. It is known that lines $M A$ and $M D$ are perpendicular to each other and that the perimeter of rectangle $A B C D$ is 24. Find its sides.
Prove that triangle $ABM$ is isosceles.

## Solution

From the equality of right triangles $ABM$ and $DCM$ (by two legs), it follows that $AM=MD$. Therefore,

$\angle MAD=\angle MDA=45^{\circ}$. Consequently, $AB=BM=MC$. Since $AB+BM+MC=12$, then $AB=4$, and $BC=8$.

## Answer

4 and 8.