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Question:

Let A and B be two regular polygons having a and b sides, respectively. If b = 2a and each interior angle of B is 3/2 times each interior angle of A, then each interior angle, in degrees, of a regular polygon with a + b sides is

Explanation:

Sum of the interior angles of a n-sided Polygon = (n - 2) × 180

Measure of each interior angle = $\frac{(n - 2)}{n}$ × 180°

So, Interior angle of the polygon with side a = $\frac{(a - 2)}{a}$ × 180°

Interior angle of the polygon with side b = $\frac{(2 a - 2)}{2 a}$ × 180°

It is given that interior angle of side b is 3/2 times to that of the polygon with side A

⇒ $\frac{(2 a - 2)}{2 a}$ × 180° = $\frac{3}{2} [\frac{(a - 2)}{a} \times 180 °]$

⇒ (2a - 2) × 180 = (3a - 6) × 180

⇒ 2a - 2 = 3a - 6

⇒ a = 4. So, b = 8

Now, (a + b) sides = 8 + 4 = 12 sides

Interior angle = $\frac{12 - 2}{12}$ × 180° = 150°

Hence, 150.

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