Question

The number of sides of two regular polygons are as 5 : 4 and the difference between their angles is $9^{\circ }$. Find the number of sides of the polygons.

Answer 1

Let n and m be the number of sides in the two regular polygon respectively.

We know that each angle of n - sided regular polygon is $\left(\frac{2n-4}{n}\right)$ right angles.

Now,

According to the question

$\frac{n}{m}=\frac{5}{4}\Rightarrow \frac{5m}{4}=n$ . . . (i)

Also,

$\left(\frac{2n-4}{n}\right){90}^{\circ }-\left(\frac{2m-4}{m}\right){90}^{\circ }=9^{\circ }$
$\Rightarrow \frac{(2n-4)m-(2m-4)n}{mn}={\left(\frac{1}{10}\right)}^{\circ }$ . . . (ii)

From (i) and (ii), we get

$\frac{(2\times \frac{5}{4}m-4)m-(2m-4)\frac{5}{4}m}{\frac{5}{4}{m}^2}=\frac{1}{10}\Rightarrow \frac{(10m-16)-(10m-20)}{5m}=\frac{1}{10}\Rightarrow \frac{4}{m}=\frac{1}{m}\Rightarrow m=8$

From (i)

$n=\frac{5}{4}m=10$

Thus,

n = 10, m = 8