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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,
h2 + x2 = 4 (1)

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

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

Hence 4x = 7 implies 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?

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

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