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# geometry

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In a certain regular polygon, the measure of each interior angle is $$10$$ times the measure of each exterior angle. Find the number of sides in this regular polygon.

Jan 10, 2022

#1
+26
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Let the measure of an exterior angle of the polygon be $$x$$. Then $$x + 10x = 180^\circ$$, so $$x = \left(\frac{180}{11}\right)^\circ$$. The measure of an exterior angle of a regular polygon with $$\ell$$ sides is equal to $$\left(\frac{360}{\ell}\right)^\circ$$. Thus, $$\left(\frac{360}{\ell}\right)^\circ = \left(\frac{180}{11}\right)^\circ$$, so the polygon has $$\boxed{22}$$ sides.

Jan 10, 2022
#2
+36455
+1

Sum of interior anges =   ( n-2) * 180

now divide by n to find the measure of each interior angle    (n-2)* 180/n

the exterior angles are supplementary   ( and exterior is 10 times bigger)

10*  ( 180 - (n-2)*180/n)   =   (n-2)*180/n

1800 - 10(n-2)*180/n = (n-2)*180/n

1800 = 11 (n-2)*180/n

1800n = (11n-22) * 180

-180n = -3960

n = 22 sides

Jan 10, 2022