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Question

The ratio of number of sides of two regular polygons are as 5:4 and the difference of there exterior angles is 9o. Find the number of sides of both the polygon.

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Solution

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 360o.
So, the exterior angle of the regular 5n-sided polygon is 360o5n.
Similarly, the exterior angle of the regular 4n-sided polygon is 360o4n.
The difference between the corresponding exterior angles is 9o.
360o4n360o5n=9o

14n15n=9o360o

5n4n20n2=140

n20n2=140

1n=12
n=2
Number of sides =4n=2×4=8 and 5n=2×5=10.

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