AMC8 2023

AMC8 2023 · Q19

AMC8 2023 · Q19. It mainly tests Triangles (properties), Area & perimeter.

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 $\frac23$ the side length of the larger triangle. What is the ratio of the area of one trapezoid to the area of the inner triangle?

一个等边三角形被放置在大等边三角形内部，使得它们之间的区域可以分成三个全等的梯形，如下图所示。内三角形的边长是大三角形的 $\frac23$。一个梯形的面积与内三角形面积的比是多少？

(A) 1 : 3 1 : 3

(B) 3 : 8 3 : 8

(D) 7 : 16 7 : 16

(E) 4 : 9 4 : 9

Answer

Correct choice: (C)

正确答案：（C）

Solution

All equilateral triangles are similar. For the outer equilateral triangle to the inner equilateral triangle, since their side-length ratio is $\frac32,$ their area ratio is $\left(\frac32\right)^2=\frac94.$ It follows that the area ratio of three trapezoids to the inner equilateral triangle is $\frac94-1=\frac54,$ so the area ratio of one trapezoid to the inner equilateral triangle is \[\frac54\cdot\frac13=\frac{5}{12}=\boxed{\textbf{(C) } 5 : 12}.\]

所有等边三角形都是相似的。对于外等边三角形到内等边三角形，由于它们的边长比是 $\frac32$，面积比是 $\left(\frac32\right)^2=\frac94$。因此，三个梯形与内等边三角形的面积比是 $\frac94-1=\frac54$，所以一个梯形与内等边三角形的面积比是 \[\frac54\cdot\frac13=\frac{5}{12}=\boxed{\textbf{(C) } 5 : 12}.\]

Topics