mathschallenge.net

Problem

An isosceles trapezium ABCD is placed inside a semicircle such that they share the same base, AD = 4, and the lengths AB = DC = 1 are chords.

Find the length BC.

Solution

Consider the following diagram.

By applying the Pythagorean theorem to right-angle triangles on the right,
h² + x² = 4 (1)

In the left hand triangle,
(2 x)² + h² = 1
\4 4x + x² + h² = 1 (2)

By substituting (1) into (2), we get,
4 4x + 4 = 1

Hence 4x = 7 x = 7/4.

So BC = 2x = 7/2.

Given the side length of the trapezium, a, and the diameter of the semicircle, d, can you find an expression for the length of the top?