¶ 2011 AMC 12B Problem 16
Problem:
Rhombus has side length 2 and . Region consists of all points inside the rhombus that are closer to vertex than any of the other three vertices. What is the area of ?
Answer Choices:
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E.
Solution:
Let and be the midpoints of and , respectively. The line drawn perpendicular to through divides the rhombus into two regions: points that are closer to vertex than , and points that are closer to vertex than . Let be the intersection of this line with diagonal . Similarly, let point be the intersection of the diagonal with the perpendicular to drawn from . Then the desired region is the pentagon .
Note that is a triangle with . Hence the area of is . Both and are congruent to , so they have the same areas. Also \
, so is an equilateral triangle. In fact, the altitude from to divides into two triangles, each congruent to . Hence the area of is .
The problems on this page are the property of the MAA's American Mathematics Competitions