Question

The number of sides of two regular polygons is 5:4 and the difference between their angles is 9^°. Find the number of sides of the polygons.

Answer 1

Let the number of sides in the first polygon be 5x and the number of sides in the second polygon be 4x.

We know:

Angle of an n-sided regular polygon \(=\left(\frac{n-2}{n}\right) 180^{\circ}\)

Thus, we have:

Angle of the first polygon \(=\left(\frac{5 x-2}{5 x}\right) 180^{\circ}\)

Angle of the second polygon \(=\left(\frac{4 x-2}{4 x}\right) 180^{\circ}\)

Now,

\(\left(\frac{5 x-2}{5 x}\right) 180-\left(\frac{4 x-2}{4 x}\right) 180=9 \)

\(\Rightarrow 180\left(\frac{4(5 x-2)-5(4 x-2)}{20 x}\right)=9\)

\(\Rightarrow \frac{20 x-8-20 x+10}{20 x}=\frac{9}{180}\)

\( \Rightarrow \frac{2}{20 x}=\frac{1}{20} \)

\( \Rightarrow \frac{2}{x}=1 \)

\(\Rightarrow x=2\)

Thus, we have:

Number of sides in the first polygon \(=5 x=10\)

Number of sides in the second polygon \(=4 x=8\)

Answer 2

Suppose the number of sides in the first polygon be 5x and

The number of sides in the second polygon be 4x.

As we know that, angle of an n-sided regular polygon = [(n - 2)/n]π radian

The angle of the first polygon = [(5x - 2)/5x]180^°

The angle of the second polygon = [(4x - 1)/4x]180^°

Hence,

[(5x - 2)/5x]180^° – [(4x - 1)/4x]180^° = 9

180^°[(4(5x - 2) – 5(4x - 2))/20x] = 9

Now, upon cross-multiplication we get,

(20x – 8 – 20x + 10)/20x = 9/180

2/20x = 1/20

2/x = 1

x = 2

∴Number of sides in the first polygon = 5x = 5(2) = 10

Thus, the number of sides in the second polygon = 4x = 4(2) = 8