In a circle of radius $R$, two mutually perpendicular diameters are given. A point on the circumference is projected onto these diameters. Find the distance between the projections of the point.

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The diagonals of a rectangle are equal.

## Solution

Let $M$ be the given point, $P$ and $Q$ be its projections on the given diameters, and $O$ be the center of the circle. Then $M P O Q$ is a rectangle. Since the diagonals of a rectangle are equal, $P Q = O M = R$.

![](https://cdn.mathpix.com/cropped/2024_05_06_02603d12f3437ea52c37g-10.jpg?height=440&width=446&top_left_y=1321&top_left_x=815)

## Answer

$R$.