Question

The ratio of the sides of two regular polygons is 1 : 2 and their interior angles is 3 : 4, then the number of sides in each polygon is

• A. 5, 10
• B. 9, 12
• C. 10, 5
• D. 5, 12

Answer 1

The correct option is A 5, 10

$\Rightarrow$ The ratio of the sides of two polygon is 1 : 2.

$\Rightarrow$ Let the polygon A have n sides & polygon B have 2n sides.

$\Rightarrow$ The sum of the interior angles of A is (n - 2)$\times$180${}^{\circ }$ = 180${}^{\circ }$n - 360${}^{\circ }$

$\Rightarrow$ So each interior angle,

$\Rightarrow$ $\frac{{180}^{\circ }n-{360}^{\circ }}{n}$ --- (1)

$\Rightarrow$ Sum of interior angles of B is (2n - 2)$\times {180}^{\circ }$ = 360${}^{\circ }$n - 360${}^{\circ }$.

$\Rightarrow$ $\frac{{360}^{\circ }n-{360}^{\circ }}{2n}$ --- (2)

$\Rightarrow$ Now the ratio of the interior angles of A and B.

$\Rightarrow$ $\frac{{180}^{\circ }n-{360}^{\circ }}{n}$ $:$ $\frac{360n^{\circ }-{360}^{\circ }}{2n}$ $::$ $\frac{3}{4}$ ---- [From (1) and (2)]

$\Rightarrow$ $\frac{{360}^{\circ }n-{720}^{\circ }}{{360}^{\circ }n-{360}^{\circ }}$ = $\frac{3}{4}$

$\Rightarrow$ $\frac{{360}^{\circ }(n-2)}{{360}^{\circ }n(n-1)}$ = $\frac{3}{4}$

$\Rightarrow$ $4n-8=3n-3$

$\therefore$ $n=5$ and $2n=10$

Thus, the number of sides of each polygon is 5 and 10.