Question

一个等边三角形放在一个更大的等边三角形内部，使得它们之间的区域可分成三个全等的梯形，如下图所示。内三角形边长是大三角形边长的 $\frac{2}{3}$ 。一个梯形的面积与内三角形的面积之比是多少？

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 $\frac{2}{3}$ the side length of the larger triangle. What is the ratio of the area of one trapezoid to the area of the inner triangle?

$1 : 3$

$3 : 8$

$5 : 12$

$7 : 16$

$4 : 9$

答案：C

知识点：面积比相似

难度评级：1410

视频讲解：

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文字解答：

内三角形边长是大三角形的 $\frac{2}{3}$ ，所以面积是大三角形的 $\left(\frac{2}{3}\right)^2 = \frac{4}{9}$ 。

三个梯形总面积是大三角形的 $1 - \frac{4}{9} = \frac{5}{9}$ 。

一个梯形面积是大三角形的 $\frac{5}{9} \div 3 = \frac{5}{27}$ 。

它与内三角形的面积比为 $\dfrac{\frac{5}{27}}{\frac{4}{9}} = \dfrac{5}{27} \cdot \dfrac{9}{4} = \dfrac{5}{12}.$

正确答案是 C。

Since the inner triangle's side length is $\frac{2}{3}$ the side length of the outer triangle, its area is $\left(\frac{2}{3}\right)^2 = \frac{4}{9}$ the area of the outer triangle.

This means that the three trapezoids are $1 - \frac{4}{9} = \frac{5}{9}$ the area of the outer triangle.

Therefore, one trapezoid is $\frac{5}{9} \div 3 = \frac{5}{27}$ the area of the outer triangle.

This makes the ratio of the areas of one trapezoid and the inner triangle $\dfrac{\frac{5}{27}}{\frac{4}{9}} = \dfrac{5}{27} \cdot \dfrac{9}{4} = \dfrac{5}{12}.$

Thus, C is the correct answer.

2023 AMC 8 第 19 题

答案：C

难度评级：1410

视频讲解：

文字解答：

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