Question 1012114
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In a regular polygon, the exterior angle is one-eighth of an interior angle. How many sides has the polygon?
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<pre>
First, let us find the interior angle.
Let x be the exterior angle. Then the interior angle is 8x.

Their sum is 180°. It gives you an equation

x + 8x = 180,   or   9x = 180,   or x = {{{180/9}}} = 20°.

Thus the interior angle {{{alpha}}} = {{{8*x}}} = {{{8*20}}} = 160°.

Now use the formula for the sum of interior angles of n-sided regular polygon.
It gives you an equation to determine n:

{{{n*alpha}}} = {{{180*(n-2)}}},   or

n*160 = 180*(n-2).

Simplify and solve it:

160n = 180n - 360  ----->   20n = 360  ----->   n = {{{360/20}}} = 18.

<U>Answer</U>. n = 18.
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