Discussion

The number of sides of two regular polygons are in the ratio 5:4 and the difference between their interior angles is 9°. Find the number of sides.

Answer: A

Correct answer: 10, 8

Let the polygons have \(5x\) and \(4x\) sides. Each interior angle of a regular \(n\)-gon is

\[\frac{(n-2)180°}{n} = 180° - \frac{360°}{n}\]

The polygon with more sides has the larger interior angle, so the difference is

\[\left(180° - \frac{360°}{5x}\right) - \left(180° - \frac{360°}{4x}\right) = \frac{360°}{4x} - \frac{360°}{5x} = \frac{360°}{20x} = \frac{18°}{x}\]

Setting this equal to \(9°\) gives \(x = 2\), so the polygons have 10 and 8 sides.

Checking: a regular decagon has interior angle \(144°\) and a regular octagon \(135°\), a difference of \(9°\).

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