Problem:
Quadrilateral is a parallelogram, and is the midpoint of the side . Let be the intersection of lines and . What is the ratio of the area of quadrilateral to the area of ?
Answer Choices:
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Solution:
Triangles and are similar by Angle-Angle. The ratio of corresponding sides is , so the ratio of their areas is . Furthermore, consider and . Because and the heights of the two triangles corresponding to bases and are the same, their areas are in a ratio. Finally, observe that the area of is the area of parallelogram . Therefore, using the area of as unit, the area of is 4 , the area of is 2 , the area of quadrilateral is , and the area of quadrilateral is . The requested ratio of the areas of quadrilateral and is .
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Note: The answer can also be obtained by assuming that the parallelogram is a square with vertices , and .
The problems on this page are the property of the MAA's American Mathematics Competitions