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
Answers
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.