Problem:
A triangle with vertices , and is reflected about the line to create a second triangle. What is the area of the union of the two triangles?
Answer Choices:
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Solution:
The reflected triangle has vertices , and . The point is on the line segment from to . The line segment from to contains the point , which must be on both triangles, and by symmetry the point is on the line segment from to . Therefore the union of the two triangles is also the union of two congruent triangles with disjoint interiors, each having the line segment from to as a base. The altitude of one of the two triangles is the distance from the line to the point , which is . Hence the union of the triangles has area .
The problems on this page are the property of the MAA's American Mathematics Competitions