Mock AIME 4 2006-2007 Problems/Problem 4
Problem
Points ,
, and
are on the circumference of a unit circle so that the measure of
is
, the measure of
is
, and the measure of
is
. The area of the triangular shape bounded by
and line segments
and
can be written in the form
, where
and
are relatively prime positive integers. Find
.
Solution
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Let the center of the circle be . The area of the desired region is easily seen to be that of sector
plus the area of triangle
minus the area of triangle
. Using the area formula
to compute the areas of the two triangles, this is
, so the answer is
.