Question

The ratio between the number of sides of two regular polygons is $3:4$ and the ratio between the sum of their interior angles is $2:3$. Find the number of sides in each polygon

• A. $6$ and $8$
• B. $7$ and $8$
• C. $3$ and $4$
• D. $12$ and $16$

Answer 1

Answer: (A)

$6$ and $8$

The ratio between the number of sides of two regular polygons is $3:4$ and the ratio between the sum of their interior angles is $2:3$.
Let number of sides of 1st polygon is $n_1$ and 2nd polygon is $n_2$.
So,
$\frac{n_1}{n_2}=\frac{3}{4}$ and $\frac{{180}^o(n_1-2)}{{180}^o(n_2-2)}=\frac{2}{3}$
$=>n_1=\frac{3}{4}{n}_2$ and $=>3n_1-6=2n_2-4$
$=>n_1=\frac{3n_2}{4}$ and $=>n_1=\frac{2(n_2+1)}{3}$
Then,
$\frac{3n_2}{4}=\frac{2(n_2+1)}{3}$
$=>9n_2-8n_2=8$
$=>n_2=8$
Now,
$=>n_1=\frac{3n_2}{4}=n_1=\frac{3\times 8}{4}=6$