Given a quadrilateral, the sum of the diagonals of which is 18. Find the perimeter of the quadrilateral with vertices at the midpoints of the sides of the given one.

#
Draw a diagonal of the quadrilateral and use the theorem about the midline of a triangle.

## Solution

Let $K, L, M$ and $N$ be the midpoints of the sides $A B, B C, C D$ and $A D$ of the quadrilateral $A B C D$. Then $K L$ and $M N$ are the midlines of triangles $A B C$ and $A C D$ with the common side $A C$. Therefore,

$$
K L=M N=\frac{1}{2} A C
$$

Similarly, we can prove that

$$
K N=L M=\frac{1}{2} B D
$$

Therefore,

$$
K L+L M+M N+K N=(K L+M N)+(K N+L M)=A C+B D=18
$$

## Answer

18.