How to approach and obtain answers when calculating ratios of graphical areas?

  Today, I would like to share a 2023 AMC8 problem with you that involves calculating ratios of graphical areas.

  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 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?

  Essential knowledge: In two similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding side lengths.

  First consider whether the area can be directly calculated.

  Neither the side lengths nor the height of either the larger or smaller triangle are known, and the height of the shaded trapezoid area is also unknown. If we were to assume that the side length is x, the entire calculation would become very complicated, which is not what we want to see. Furthermore, our ultimate goal is to calculate ratios, not specific areas. Therefore, we abandon the idea of directly calculating the area here.

  Consider from the properties of similar triangles.

  As described in the essential knowledge section: in two similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding side lengths. We know that the ratio of the side lengths of the inner triangle to the side lengths of the outer triangle is 2:3. Therefore, the ratio of the area of the inner triangle to the area of the outer triangle is 4:9.

Use fraction concepts to understand ratio relationships.

  The ratio of the area of the inner triangle to the area of the outer triangle is 4:9. We can think of the area of the inner triangle as 4 parts and the area of the outer triangle as 9 parts, so the sum of the areas of the three trapezoids is 5 parts, and the area of each trapezoid is 5/3 parts. Therefore, the ratio of the total area of the trapezoids to the area of the inner triangle is 5/3 : 4 ,which can be simplified to 5:12.