Question

two regular polygon are such that the ratio between their number of side is 1:3 and ratio of number of interior angle 3:4​

Answer 1

Note:-

• Regular polygon - A polygon whose all sides are equal is called regular polygon.

• All the interior angles of a regular polygon are equal.

• All the exterior angles of a regular polygon are equal.

• Sum of all interior angles of a polygon with n sides is equal to (n-2)×180° .

• Measure of an interior angle of a regular polygon of n sides is equal to (n-2)×180°/n .

Answer:

6 and 18

Solution:

Let 1st polygon and 2nd polygon has n and n' sides respectively .

According to the question,

The ratio of sides is 1:3 .

Thus,

=> n:n' = 1:3

=> n/n' = 1/3

=> 3×n = 1×n'

=> n' = 3n

Now,

Measure of an interior angle of 1st polygon

= (n-2)×180°/n

Also,

Measure of an interior angle of 2nd polygon

= (n'-2)×180°/n'

Now,

According to the question,

Ratio of interior angle of both the polygon is 3:4 .

Thus,

(n-2)×180°/n

=> –––·––––––– = 3/4

(n'-2)×180°/n'

(n-2)/n

=> –––––– = 3/4

(n'-2)/n'

=> n'×(n-2) / n×(n'-2) = 3/4

=> 4×n'×(n-2) = 3×n×(n'-2)

=> 4×3n×(n-2) = 3×n×(3n-2). { n' = 3n }

=> 4(n-2) = (3n-2)

=> 4n - 8 = 3n - 2

=> 4n - 3n = 8 - 2

=> n = 6

Thus,

n' = 3×6 = 18

Hence,

The polygons under consideration have 6 and 18 sides.