AMC 8 2023 Problem 19
AMC8 2023 年第 19 题
三角形★★★☆☆
An equilateral triangle is placed inside a larger equilateral triangle so that the region between them can be divided into three congruent trapezoids, as shown below. The side length of the inner triangle is the side length of the larger triangle. What is the ratio of the area of one trapezoid to the area of the inner triangle?
如图所示，一个等边三角形在另一个更大的等边三角形内部，它们之间的区域可以分成三个全等的梯形．内部三角形的边长是大三角形边长的，问一个梯形的面积与内部三角形的面积之比是多少？
- A.1: 3
- B.3: 8
- C.5: 12
- D.7: 16
- E.4: 9
正确答案 · Correct Answer
C
解析 · Solution
Solution 1
All equilateral triangles are similar. For the outer equilateral triangle to the inner equilateral triangle, since their side-length ratio is their area ratio is It follows that the area ratio of three trapezoids to the inner equilateral triangle is so the area ratio of one trapezoid to the inner equilateral triangle is
Subtracting the larger equilateral triangle from the smaller one yields the sum of the three trapezoids. Since the ratio of the side lengths of the larger to the smaller one is , we can set the side lengths as 3 and 2, respectively. So, the sum of the trapezoids is . We are also told that the three trapezoids are congruent, thus the area of each of them is . Hence, the ratio is .
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