AMC 8 2014 Problem 14
AMC8 2014 年第 14 题
四边形与多边形★★★☆☆
Rectangle ABCD and right triangle DCE have the same area. They are joined to form a trapezoid, as shown. What is DE ?
矩形 ABCD 和直角三角形 DCE 的面积相同。他们拼起来形成了一个梯形，如下图所示。那么 DE 的长度是多少？
- A.12
- B.13
- C.14
- D.15
- E.16
正确答案 · Correct Answer
B
解析 · Solution
Solution 1
The area of is . The area of is , which also must be equal to the area of , which, since DC=5, must in turn equal . Through transitivity, then, , and CE=12. Then, using the Pythagorean Theorem, you should be able to figure out that is a 5-12-13 triangle, so DE=13, or.
The area of the rectangle is 5×6=30. Since the parallel line pairs are identical, DC=5. Let be . is the area of the right triangle. Solving for , we get x=12. According to the Pythagorean Theorem, we have a 5-12-13 triangle. So, the hypotenuse has to be.
This problem can be solved with the Pythagorean Theorem ( ). We know AB = DC, so DC = 5. is twice the length of , so CE = 12. . . . 25 + 144 = 169. 169 has a square root of 13, so the hypotenuse or is 13. The answer is. —
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