Problem:
A triangle is partitioned into three triangles and a quadrilateral by drawing two lines from vertices to their opposite sides. The areas of the three triangles are , and , as shown. What is the area of the shaded quadrilateral?
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Answer Choices:
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Solution:
Partition the quadrilateral into two triangles and let the areas of the triangles be and as shown. Then the required area is .
Let and , respectively, be the bases of the triangles with areas and , as indicated. If two triangles have the same altitude, then the ratio of their areas is the same as the ratio of their bases. Thus
Similarly,
Thus
From this we obtain
and it follows that .
The problems on this page are the property of the MAA's American Mathematics Competitions