Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.5 AIME Find the answer

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. Spacing and $ signs are ignored.

Solution

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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