2005 AMC 10B Problems/Problem 14
Problem
Equilateral has side length , is the midpoint of , and is the midpoint of . What is the area of ?
Solution
Solution 1
The area of a circle can be given by . because it is the midpoint of a side, and because it is twice the length of . Each angle of an equilateral triangle is so . The area is .
Solution 2
In order to calculate the area of , we can use the formula , where is the base. We already know that , so the formula now becomes . We can drop verticals down from and to points and , respectively. We can see that . Now, we establish the relationship that . We are given that , and is the midpoint of , so . Because is a triangle and the ratio of the sides opposite the angles are is . Plugging those numbers in, we have . Cross-multiplying, we see that Since is the height , the area is .
See Also
2005 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
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