Question : If one of the interior angles of a regular polygon is $\frac{15}{16}$ times of one of the interior angles of a regular decagon, then find the number of diagonals of the polygon.
Option 1: 20
Option 2: 14
Option 3: 2
Option 4: 35
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Correct Answer: 20
Solution :
Interior angle of a regular polygon = $\frac{(n-2) \times180^{\circ}}{n}$
Interior angle of a regular decagon = $\frac{(10-2) \times180^{\circ}}{10}$ = $144^{\circ}$
Interior angle of a regular polygon = $\frac{15}{16}×144^{\circ}$ = $135^{\circ}$
⇒ $\frac{(n-2) \times 180^{\circ}}{n}$ = $135^{\circ}$
⇒ $180n - 135n$ = $360$
⇒ $n$ = $\frac{360}{45}$ = 8
Number of diagonals = $\frac{n(n-3)}{2}$ = $\frac{8(8-3)}{2}$ = 20
Hence, the correct answer is 20.
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