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# GEOMETRY

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

Jun 19, 2018

#1
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The interior angle of a regular polygon can be determined with the following formula:

$$\frac{180(n-2)}{n}$$ where n represents the number of sides the polygon has

The measure of each exterior angle of a regular polygon can be determined using the following formula:

$$\frac{360}{n}$$ where n represents the number of sides the polygon has

According to the problem, "the measure of each interior angle is four times the measure of each exterior angle," so we can create an equation for this like the following:

 $$\frac{180(n-2)}{n}=4\left(\frac{360}{n}\right)$$ Simplify both sides of the equation. $$\frac{180n-360}{n}=\frac{1440}{n}$$ Because the denominators are equal, simply set the numerators equal to each other and solve. $$180n-360=1440$$ Add 360 to both sides. $$180n=1800$$ Divide by 180 $$n=10\text{ sides}$$
Jun 20, 2018
#2
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I like the second way better.

x+4x=180. 5x=180. x=36.

Then using knowledge that the number of sides of a regular polygon is equal to 360/exterior angle measure, we see that 360/36=10, so there are 10 sides

Jun 20, 2018