Problem:
Let be the kite formed by joining two right triangles with legs 1 and along a common hypotenuse. Eight copies of are used to form the polygon shown below. What is the area of triangle ?
Answer Choices:
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Solution:
Let points , and be labeled as shown. Points , and are collinear, and the line through them is perpendicular to line . Furthermore, the distance from to line is equal to the distance from to line , which in turn is equal to . The area of is thus .
To compute , observe that is the longer leg of a triangle with hypotenuse . This implies that , from which . To compute , observe that in the same right triangle as before, . This implies that .\
Therefore the area of is .
The problems on this page are the property of the MAA's American Mathematics Competitions