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# The ratio of the number of sides of two regular polygons is 5 : 4 and the difference of their exterior angles is ${9^ \circ }$ . Find the number of sides of both the polygons.

Last updated date: 22nd Feb 2024
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Hint: Considering n be the greatest common divisor(GCD).Then one polygon has 5n sides,while other has 4n sides and we have to calculate the exterior angle of polygon of 5n-sides and 4n-sides and find their differences

Let n be the greatest common divisor (GCD) of the numbers under the question.
Then one polygon has 5n sides, while other has 4n sides
It is well known fact that the sum of exterior angles of each polygon is ${360^ \circ }$
So, the exterior angle of the regular 5n-sided polygon is $\dfrac{{{{360}^ \circ }}}{{5n}}$
Similarly, the exterior angle of the regular 4n-sided polygon is $\dfrac{{{{360}^ \circ }}}{{4n}}$
According to question it is given that difference between the corresponding exterior angles is ${9^ \circ }$
$\Rightarrow \dfrac{{{{360}^ \circ }}}{{4n}} - \dfrac{{{{360}^ \circ }}}{{5n}} = {9^ \circ }$
$\Rightarrow \dfrac{{5n - 4n}}{{20{n^2}}} = \dfrac{9}{{360}} = \dfrac{1}{{40}}$
$\Rightarrow 20n = 40$
$\Rightarrow n = 2$
So, number of sides in one polygon = $5n = 5 \times 2 = 10$
And number of sides in another polygon $= 4n = 4 \times 2 = 8$