2009 AMC 12A Problems/Problem 19
Problem
Andrea inscribed a circle inside a regular pentagon, circumscribed a circle around the pentagon, and calculated the area of the region between the two circles. Bethany did the same with a regular heptagon (7 sides). The areas of the two regions were and
, respectively. Each polygon had a side length of
. Which of the following is true?
Solution
In any regular polygon with side length , consider the isosceles triangle formed by the center of the polygon
and two consecutive vertices
and
. We are given that
. Obviously
, where
is the radius of the circumcircle. Let
be the midpoint of
. Then
, and
, where
is the radius of the incircle.
Applying the Pythagorean theorem on the triangle , we get that
.
Then the area between the circumcircle and the incircle can be computed as .
Hence ,
, and therefore
.
See Also
2009 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 18 |
Followed by Problem 20 |
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All AMC 12 Problems and Solutions |