AMC 8 2002 Problem 16
AMC8 2002 年第 16 题
三角形★★★☆☆
Right isosceles triangles are constructed on the sides of a 3-4-5 right triangle, as shown. A capital letter represents the area of each triangle. Which one of the following is true?
如下图所示，在 3-4-5 直角三角形的三条边上作出三个等腰直角三角形。大写字母表示每个三角形的面积。下面哪个是正确的？
- A.X+Z=W+Y
- B.W+X=Z
- C.3X+4Y=5Z
- D.
- E.X+Y=Z
正确答案 · Correct Answer
E
解析 · Solution
Solution 1
The area of a right triangle can be found by using the legs of the triangle as the base and height. In the three isosceles triangles, the length of their second leg is the same as the side that is connected to the 3-4-5 triangle. $$$$$$$$ Plugging into the answer choices, the only one that works is X+Y=Z.
Looking at the diagram, we notice that three right isosceles triangles on one right triangle reminds us of the Pythagorean theorem, since each right isosceles triangle is actually half of a square. Each square's area represents a side length squared, so the squares on the legs of the right triangle adds to the square on the hypotenuse. This gives 2X+2Y=2Z. Then, dividing by 2 we get X+Y=Z, which is one of the answer choices. Since there can only be one correct answer, and there is already one, we see that the answer must be X+Y=Z.
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