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

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

Solution:

Let the first polygon have 5x sides and the second polygon have 5x sides.

In the second polygon, the number of sides should be 4x.

Angle of an n-sided regular polygon = [(n-2)/n] radian, we know.

Therefore, the angle of the first polygon = [(5x-2)/5x] 180^o

Similarly, the angle of the second polygon = [(4x-1)/4x] 180^o

Thus, we can write:

$\left[ \left( 5x-2 \right)/5x \right]\text{ }{{180}^{o}}~-\left[ \left( 4x-1 \right)/4x \right]\text{ }{{180}^{o}}~=\text{ }9$

${{180}^{o}}~\left[ \left( 4\left( 5x-2 \right)-5\left( 4x-2 \right) \right)/20x \right]\text{ }=\text{ }9$

Upon cross multiplying we get,

$\left( 20x-8-20x\text{ }+\text{ }10 \right)/20x\text{ }=\text{ }9/180$

$2/20x\text{ }=\text{ }1/20$

$2/x\text{ }=\text{ }1$

$x\text{ }=\text{ }2$

Terefore, the number of sides in the first polygon = 5x = 5(2) = 10

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

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