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2012 AMC 12A Problem 10

Problem 10 of 25EasierGeometry

A triangle has area 30,30, one side of length 10,10, and the median to that side of length 9.9. Let θ\theta be the acute angle formed by that side and the median. What is sin⁡θ?\sin\theta?

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Solution

The median divides the triangle into two triangles of equal area 15.15. One of them has the two sides of length 55 (half the base) and 99 (the median) meeting at angle θ.\theta. Its area is 12⋅5⋅9sin⁡θ=15,\tfrac12 \cdot 5 \cdot 9 \sin\theta = 15, so sin⁡θ=2⋅155⋅9=23.\sin\theta = \dfrac{2 \cdot 15}{5 \cdot 9} = \dfrac{2}{3}. Thus, the correct answer is D.
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Tagged: triangle area · median (geometry) · trigonometry

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