Problem:
A triangle has area 30 , one side of length 10 , and the median to that side of length 9 . Let be the acute angle formed by that side and the median. What is ?
Answer Choices:
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Solution:
The area of a triangle equals one half the product of two sides and the sine of the included angle. Because the median divides the base in half, it partitions the triangle in two triangles with equal areas. Thus , and .
OR
The altitude to the base forms a right triangle with the median as its hypotenuse, and thus . Hence the area of the original triangle is , so .
The problems on this page are the property of the MAA's American Mathematics Competitions