2012 AMC 12A Problems/Problem 10
Problem
A triangle has area , one side of length , and the median to that side of length . Let be the acute angle formed by that side and the median. What is ?
Solution
Solution 1
is the side of length , and is the median of length . The altitude of to is because the 0.5(altitude)(base)=Area of the triangle. is . To find , just use opposite over hypotenuse with the right triangle . This is equal to .
Solution 2
It is a well known fact that a median divides the area of a triangle into two smaller triangles of equal area. Therefore, the area of in the above figure. Expressing the area in terms of , . Solving for gives . .
Solution 3
The area of a triangle with sides and angle between them is Therefore, as two angles along the same line must be supplementary. This simplifies to
See Also
2012 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 9 |
Followed by Problem 11 |
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All AMC 12 Problems and Solutions |
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