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.
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?