Problem:
Two sides of a triangle have lengths and . The length of the altitude to the third side is the average of the lengths of the altitudes to the two given sides. How long is the third side?
Answer Choices:
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Solution:
Denote the length of the third side as , and the altitudes to the sides of lengths and as and , respectively. Then twice the area of the triangle is . This implies that , so
Therefore , and .
The problems on this page are the property of the MAA's American Mathematics Competitions