[ Varignon Parallelogram ] [ Area of a Quadrilateral ]

The segments connecting the midpoints of opposite sides of a convex quadrilateral are equal to each other. Find the area of the quadrilateral if its diagonals are 8 and 12.
Let $K, L, M$ and $N$ be the midpoints of the sides $A B, B C, C D$ and $A D$ of the given convex quadrilateral $A B C D$. Since $K L$ and $M N$ are the midlines of triangles $A B C$ and $A D C$, we have $K L \| M N$ and $K L = M N$, thus quadrilateral $K L M N$ is a parallelogram. Since its diagonals $K M$ and $L N$ are equal, $K L M N$ is a rectangle. The sides of rectangle $K L M N$ are parallel to the diagonals $A C$ and $B D$ of quadrilateral $A B C D$, so the diagonals of quadrilateral $A B C D$ are perpendicular to each other. Therefore,

$$
S(A B C D)=\frac{1}{2} A C \cdot B D=\frac{1}{2} \cdot 8 \cdot 12=48
$$

## Answer

48.