## Isosceles Trapezium

#### 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,`h`^{2} + `x`^{2} = 4 (1)

In the left hand triangle,

(2 `x`)^{2} + `h`^{2} = 1

\4 4`x` + `x`^{2} + `h`^{2} = 1 (2)

By substituting (1) into (2), we get,

4 4`x` + 4 = 1

Hence 4`x` = 7 `x` = 7/4.

So `BC` = 2`x` = 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?

Problem ID: 79 (May 2002) Difficulty: 3 Star