SOLUTION: the measure of an interior angle of a regular polygon is 144. finsd the apothem if one side of the polygon measures 5 units

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Question 881564: the measure of an interior angle of a regular polygon is 144. finsd the apothem if one side of the polygon measures 5 units
Answer by Theo(13342) (Show Source):
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interior angle of the polygon is equal to 144 degrees.
exterior angle and central angle of the polygon is equal to 180 - 144 degrees which is equal to 36 degrees.
you have enough to find the apothem of the polygon.
eacj each adjacent vertex of the polygon will form an isosceles triangle.
the vertex of the triangle will be the central angle of the polygon which is equal to 36 degrees.
each base angle of the isosceles triangle will be 1/2 the number of degrees of the interior angle which will be equal to 72 degrees.
the apothem is a vertical line from the central angle connecting to and perpendicular to the base.
since the triangle is isosceles, this vertical line splits the central angle into 2 angles of 18 degrees each and also splits the base of the triangle into 2 equal sections of 2.5 units apiece, since the base of this triangle is equal to a side of the polygon.
the apothem of each of the right triangles formed will be found by using the tangent formula of tangent of the base angle is equal to the opposite side divided by adjacent side.
this formula becomes tan(72) = apothem / 2.5.
solve this formula for apothem to get:
apothem = 2.5 * tan(72) which is equal to 2.5 * 3.07768 which is equal to 7.6942 rounded to 4 decimal places.
that's your answer.
the link give you more information about a decagon which is what this polygon is all about.
that's determined by dividing 360 by the central or exterior angle which becomes 360 / 36 which becomes 10.
here's a reference that talks about the properties of a polygon.
http://www.mathsisfun.com/geometry/regular-polygons.html
here's a picture of what i mean: