Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.1 AIME Prove it United States

Problem:

Regular octagon CHILDREN has area $1$ . Determine the area of quadrilateral $LINE$ .

Solution

Solution:

Suppose that the side length $CH = \sqrt{2} a$ , then the area of the octagon is $((2+\sqrt{2}) a)^2 - 4 \cdot \frac{1}{2} a^2 = (4+4 \sqrt{2}) a^2$ , and the area of $LINE$ is $(\sqrt{2} a)((2+\sqrt{2}) a) = (2+2 \sqrt{2}) a^2$ , which is exactly one-half of the area of the octagon. Therefore the area of $LINE$ is $\frac{1}{2}$ .

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