[Varignon's Parallelogram] [Area of a Quadrilateral]

The diagonals of a convex quadrilateral are equal to $a$ and $b$, and the segments connecting the midpoints of opposite sides are equal to each other. Find the area of the quadrilateral.
The midpoints of the sides of a quadrilateral are the vertices of a parallelogram.

## Solution

The midpoints of the sides of any quadrilateral are the vertices of a parallelogram. In this case, this parallelogram is a rectangle because its diagonals are equal to each other. The diagonals of this quadrilateral are parallel to the sides of this rectangle. Therefore, they are perpendicular to each other. Consequently, the desired area is $\frac{a b}{2}$.

![](https://cdn.mathpix.com/cropped/2024_05_06_16ab69fab1743849dba9g-45.jpg?height=446&width=688&top_left_y=151&top_left_x=685)

## Answer

$\frac{a b}{2}$.