4. A circle with center at point $O$ is circumscribed around quadrilateral $A B C D$. The diagonals of the quadrilateral are perpendicular. Find the length of side $B C$, if the distance from point $O$ to side $A D$ is 1.
Answer: $B C=2$

Solution. Let $O E \perp A D$, then $O E=1$. Draw a line through point $A$ perpendicular to $A D$, which intersects the circle at point $M$. Then $D M$ is a diameter and $D O=O M$. Since $\angle D B A=\angle D M A$ (as inscribed angles subtending the same arc), $\angle M D A=90^{\circ}-\angle D M A$ and $\angle C A B=90^{\circ}-\angle D B A$, then $\angle C A B=\angle M D A$. Therefore, arc $A M$ is equal to arc $B C$ and $B C=A M$. $\triangle M D A$ is similar to $\triangle O D E$ (by two equal angles) with a similarity ratio of $\frac{1}{2}$, so $2 O E=A M=B C=2$.

![](https://cdn.mathpix.com/cropped/2024_05_06_1c65f54357fef1acdf6fg-2.jpg?height=754&width=754&top_left_y=1383&top_left_x=594)

## Grading Criteria.

Similarity of $\triangle O E D$ and $\triangle C B A$ is proven - 2 points.