2009 AMC 12A Problem 19

Problem 19 of 25HarderGeometry

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

A

and

B,

respectively. Each polygon had a side length of

2.

Which of the following is true?

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Solution

For a regular polygon with side length

2,

let

O

be the center,

M

the midpoint of a side, and

N

an endpoint of that side. Then

\triangle OMN

has a right angle at

M,

with

MN = 1,

OM = r

(inradius), and

ON = R

(circumradius). So

R^2 - r^2 = 1,

and the area between the circles is

\pi(R^2 - r^2) = \pi

for any number of sides. Hence

A = B.

Thus, the correct answer is C.

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