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Harvard-MIT Mathematics Tournament

Geometry Difficulty 4.0 AIME Find the answer United States

Problem:
A regular decagon $A_{0} A_{1} A_{2} \cdots A_{9}$ is given in the plane. Compute $\angle A_{0} A_{3} A_{7}$ in degrees.

Your answer

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

Solution

Solution:
$54^{\circ}$

Put the decagon in a circle. Each side subtends an arc of $360^{\circ} / 10 = 36^{\circ}$ . The inscribed angle $\angle A_{0} A_{3} A_{7}$ contains 3 segments, namely $A_{7} A_{8}$ , $A_{8} A_{9}$ , $A_{9} A_{0}$ , so the angle is $108^{\circ} / 2 = 54^{\circ}$ .

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