[ Inscribed angle subtended by a diameter ] [

The distances from one end of the diameter to the ends of a chord parallel to it are 13 and 84. Find the radius of the circle.

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The diagonals of an isosceles trapezoid are equal.

Let $AB$ be the given diameter, $CD$ be the given chord, $AD=13, AC=84$. Trapezoid $ABCD$ is isosceles, so $BD=AC$.

From the right triangle $ADB$, we find that $AB^2 = AD^2 + DB^2 = 85^2$.

![](https://cdn.mathpix.com/cropped/2024_05_06_d3cd76ad0dc53ae543bdg-30.jpg?height=457&width=509&top_left_y=203&top_left_x=777)

## Answer

$42.5$.