the number of sides of two regular polygons are in the ratio 4:5 and their interior angles are in the ratio 15:16.Find the number of sides of each polygon.
Answers
Answered by
1
Step-by-step explanation:
Let n be the Greatest Common Divisor (GCD) of the numbers under the question.
Then one polygon has 5n sides, while the other has 4n sides.
It is well known fact that the sum of exterior angles of each (convex) polygon is 360
o
.
So, the exterior angle of the regular 5n-sided polygon is
5n
360
o
.
Similarly, the exterior angle of the regular 4n-sided polygon is
4n
360
o
.
The difference between the corresponding exterior angles is 9
o
.
⇒
4n
360
o
−
5n
360
o
=9
o
⇒
4n
1
−
5n
1
=
360
o
9
o
⇒
20n
2
5n−4n
=
40
1
⇒
20n
2
n
=
40
1
⇒
n
1
=
2
1
∴ n=2
⇒ Number of sides =4n=2×4=8 and 5n=2×5=10.
Similar questions