2008 AMC 10A Problems/Problem 20
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[hide]Problem
Trapezoid has bases and and diagonals intersecting at Suppose that , , and the area of is What is the area of trapezoid ?
Solution 1
Since it follows that . Thus .
We now introduce the concept of area ratios: given two triangles that share the same height, the ratio of the areas is equal to the ratio of their bases. Since share a common altitude to , it follows that (we let denote the area of the triangle) , so . Similarly, we find and .
Therefore, the area of .
Solution 2
We may consider that trapezoid to be right, as there is nothing specifying its angles. Consider D and A right. Let the length of DA be h. Now we let A be (0,0) and we compute the x-coordinate of K from lines AC and DB. for line DB, for line AC. Solving for K, simplifying, , . Using the fact that , we solve for h. . Applying trapezoid area formula: . Thus, the area is 98 and the answer is
See also
2008 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 19 |
Followed by Problem 21 | |
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