HMMT November

Geometry Difficulty 4.2 AIME Find the answer

Regular octagon $C H I L D R E N$ has area 1. Find the area of pentagon $C H I L D$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

The pentagon $C H I L D$ is congruent to the pentagon $N E R D C$ , as their corresponding angles and sides are congruent. Moreover, the two pentagons together compose the entire octagon, so each pentagon must have area one-half of the area of the octagon, or $\frac{1}{2}$ .

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