Problem:
Three unit squares and two line segments connecting two pairs of vertices are shown. What is the area of ?
Answer Choices:
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Solution:
Place the figure on the coordinate plane with at the origin, on the positive -axis, and label the other points as shown. Then the equation of line is , and the equation of line is . Solving the simultaneous equations shows that . Therefore has base and altitude , so its area is .
Congruent right triangles and have the property that their corresponding legs are perpendicular; hence their hypotenuses are perpendicular. Therefore is a right angle and is similar to . Because and , the ratio of the area of to that of is to . The area of is , so the area of is .
The problems on this page are the property of the MAA's American Mathematics Competitions