Question

The number of sides of two regular polygons are in the ratio 2 : 3 and their interior angles are in the ratio 9 : 10. Find the number of sides of each polygon.

Answer 1

Ratio of the sides = 2 : 3.
∴ Number of sides in each polygon is 2x and 3x.

Number angle of a regular polygon of n sides = `(("n" - 2) xx 180°)/"n"`

∴ Interior angle of a regular polygon of 2x sides = `((2x - 2) xx 180°)/(2x)`

And, interior of a regular polygon of 3x sides = `((3x - 2) xx 180°)/(3x)`

Ratio of the interior angles = 9 : 10

⇒ `((2x - 2) xx 180°)/(2x) : ((3x - 2) xx 180°)/(3x)` = 9 : 10

⇒ `((2x - 2) xx 180°)/(2x) xx (3x)/((3x - 2) xx 180°) = (9)/(10)`

⇒ `((x - 1) xx 180°)/x xx (3x)/((3x - 2) xx 180°) = (9)/(10)`

⇒ `(3(x - 1))/((3x - 2)) = (9)/(10)`

⇒ `(x - 1)/(3x - 2) = (3)/(10)`

⇒ 10x - 10 = 9x - 6
⇒ x = 4
∴ Number of sides in each polygon
= 2(4) = 8 and 3(4)
= 12.