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, 2014

Geometry Difficulty 4.9 AIME Prove it United States

Problem:

In the octagon $COMPUTER$ exhibited below, all interior angles are either $90^{\circ}$ or $270^{\circ}$ and we have $CO = OM = MP = PU = UT = TE = 1$ .

Point $D$ (not to scale in the diagram) is selected on segment $RE$ so that polygons $COMPUTED$ and $CDR$ have the same area. Find $DR$ .

Solution

Solution:

The area of the octagon $COMPUTER$ is equal to $6$ . So, the area of $CDR$ must be $3$ . So, we have the equation
$\frac{1}{2} \times CD \times DR = [CDR] = 3.$
And from $CD = 3$ , we have $DR = 2$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.