## In a convex quadrilateral $A B C D$, the length of the segment connecting the midpoints of sides $A B$ and $C D$ is 1. Lines

$B C$ and $A D$ are perpendicular. Find the length of the segment connecting the midpoints of diagonals $A C$ and $B D$.
Prove that the midpoints of sides $AB$ and $CD$ and the midpoints of diagonals $AC$ and $BD$ are the vertices of a rectangle.

## Solution

Let $M$ and $N$ be the midpoints of sides $AB$ and $CD$ of quadrilateral $ABCD$, and $P$ and $Q$ be the midpoints of its diagonals $AC$ and $BD$, respectively. Then $MP$ is the midline of triangle $ABC$, and $QN$ is the midline of triangle $DBC$. Therefore,

$$
MP = \frac{1}{2} BC = QN, \quad MP \| BC \| QN
$$

Thus, quadrilateral $MPNQ$ is a parallelogram. Its adjacent sides $MP$ and $MQ$ are parallel to lines $BC$ and $AD$, respectively, so $MP \perp MQ$. Therefore, quadrilateral $MPNQ$ is a rectangle. The diagonals of a rectangle are equal, so

$$
PQ = MN = 1
$$

![](https://cdn.mathpix.com/cropped/2024_05_06_b9b3326e3456e69a696eg-12.jpg?height=378&width=760&top_left_y=434&top_left_x=655)

## Answer

1.