Question

Regular polygons A and B have number of sides in the ratio \(1:2\) and interior angles in the ratio \(3:4\).Then the number of sides of B equals

Answer 1

The correct answer is :10
Regular polygons A and B have interior angles in the ratio \(3\ratio4\).Let's denote the number of sides of polygon A as 'n' and the number of sides of
polygon B as '2n' (since the ratio of the number of sides is \(1\ratio2\)). 
The sum of the interior angles of a polygon is given by the formula \((n - 2)\times180\degree\).For a regular polygon,each interior angle is given by \(\frac{[(n - 2)\times180]}{n} degrees\). 
According to the given ratio of interior angles, we have: 
\(\frac{[(n - 2)\times180]}{n}\ratio\frac{[(2n - 2)\times180]}{(2n)}=3\ratio4\) 
Cross-multiplying and simplifying: 
4(n-2)=3(2n-2) 
Expanding and solving for 'n': 
4n-8=6n-6 
2n=2 
n=5 
So, polygon A has 5 sides, and polygon B has \(2\times5=10\) sides.

Answer 2

Let the number of sides of polygons A and B be n and 2n, respectively 
The sum of the interior angles of a polygon is given by the formula \((n - 2)\times180\degree\).For a regular polygon each interior angle is given by \(\frac{[(n - 2)\times180\degree]}{n}\)

Now,

\(\frac{\frac{(n - 2)\times180}{n}}{\frac{(2n - 2)\times180\degree}{2n}}=\frac{3}{4}\)

\(\frac{\frac{(n - 2)\times180\degree}{n}}{\frac{2(n - 1)\times180\degree}{2n}}=\frac{3}{4}\)

\(\frac{n-2}{n-1}=\frac{3}{4}\)

\(4n-8=3n-3\)
\(n=5\)
Polygon A has 5 sides, and polygon B has 10 sides.

So, the answer is 10.