Question

Determine the number of sides of a regular polygon whose external and internal angles are in ratio 5/4 : 5/2​

Answer 1

★Given :

Ratio of external and internal angles of a regular polygon = $\frac{5}{4} : \frac{5}{2}$

★To find :

Number of sides of regular polygon

★We know that,

Each interior angle of regular polygon = $\dfrac{180(n-2)}{n}$

Each exterior angle of regular polygon = $\dfrac{360}{n}$

★Solution :

$External\:angle = \frac{5}{4} \times x = \frac{5x}{4}\\ \\ \\ Interior \:angle = \frac{5}{2} \times x = \frac{5x}{2} \\ \\ \\ by\: equating\:with\:formula \\ \\ \\ \bold{External\:angle}\\ \\ \\ \frac{360}{n} = \frac{5x}{4} \\ \\ \\ by\: cross\: multiplication\\ \\ \\ 360 \times 4 = 5xn \\ \\ \\ xn = \frac{\cancel{360} \times 4}{\cancel{5}} \\ \\ \\ xn = 72 \times 4 \\ \\ \\ xn = 288 ......(1)$

$\bold{ Interior\:angle}\\ \\ \\\frac{5x}{2} = \frac{180(n-2)}{n} \\ \\ \\ by\: cross\: multiplication\\ \\ \\ 5xn = 2 \times [180(n-2)] \\ \\ \\ 5xn = 360(n-2) \\ \\ \\ xn = \frac{\cancel{360}(n-2)}{\cancel{5}} \\ \\ \\ xn = 72(n-2) \\ \\ \\ by\: putting\:value\:of\: xn\\ \\ \\ 288 = 72(n-2) \\ \\ \\ (n-2) = \frac{\cancel{288}}{\cancel{72}} \\ \\ \\ n-2 = 4\\ \\ \\ n = 6$

${\pink{\boxed{Number\:of\:sides\:of\: regular\:polygon = 6}}}$