question_answer

Two regular polygon are such that the ratio of the measures of their interior angles is 4:3 and the ratio between their number of sides is 2:1. Find the number of sides of each polygon.

Answer:

Let 2n and n be the number of sides of the regular polygon.

\[\therefore \] Their interior angles are

\[{{\left[ \frac{2(2n)-4}{2n}\times 90 \right]}^{{}^\circ }}\,\]and \[{{\left[ \frac{2n-4}{n}\times 90 \right]}^{{}^\circ }}\,\]

Since the ratio of the interior angles is 4:3.

\[\therefore \]      \[\frac{\left[ \frac{2(2n)-4}{2n}\times 90 \right]}{\left[ \frac{(2n-4)}{n}\times 90 \right]}=\frac{4}{3}\]

\[\Rightarrow \]   \[\frac{n}{2n}\times \frac{[2(2n)-4]}{[2n-4]}=\frac{4}{3}\]

\[\Rightarrow \] \[\frac{1}{2}\times \frac{4n-4}{2n-4}=\frac{4}{3}\]

\[\Rightarrow \] \[\frac{1}{2}\times \frac{4(n-1)}{2(n-2)}=\frac{4}{3}\Rightarrow \frac{n-1}{n-2}=\frac{4}{3}\]

\[\Rightarrow \] \[3(n-1)=5(n-2)\Rightarrow 3n-3=4n-8\]

\[\Rightarrow \] \[3n-4n=-8+3\]

\[\Rightarrow \] \[-n=-5\Rightarrow n=5\]

\[\therefore \]    \[2n=2\times 5=10\]

Thus, the number of sides of polygon are 10 and 5 respectively.