A segment of constant length moves in a plane such that its ends slide along the sides of a right angle.

What is the trajectory of the midpoint of this segment?
The median of a right triangle, drawn from the vertex of the right angle, is equal to half the hypotenuse.

## Solution

Let $O$ be the vertex of the given right angle, $AB$ be the given segment, and $M$ be its midpoint. Then $OM = 1/2 AB$.

Therefore, the point $M$ lies on the circle with center at point $O$ and radius equal to half of the given segment.

![](https://cdn.mathpix.com/cropped/2024_05_06_5e9166ba6ad1f1b1f21fg-25.jpg?height=480&width=483&top_left_y=0&top_left_x=113)

## Answer

A quarter of a circle.

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