Video: Identifying the Equal Proportions between Corresponding Sides in Two Similar Triangles

Using the diagram, which of the following is equal to 𝐴𝐡/𝐴𝐷? [A] 𝐴𝐡/𝐷𝐡 [B] 𝐴𝐢/𝐴𝐸 [C] 𝐴𝐢/𝐸𝐢 [D] 𝐴𝐷/𝐷𝐡 [E] 𝐴𝐸/𝐸𝐢

02:16

Video Transcript

Using the diagram, which of the following is equal to 𝐴𝐡 over 𝐴𝐷? (A) 𝐴𝐡 over 𝐷𝐡, (B) 𝐴𝐢 over 𝐴𝐸, (C) 𝐴𝐢 over 𝐸𝐢, (D) 𝐴𝐷 over 𝐷𝐡, or (E) 𝐴𝐸 over 𝐸𝐢.

In our diagram, line segment 𝐸𝐷 is parallel to line segment 𝐢𝐡. Because of that, we have two similar triangles. We can say that triangle 𝐴𝐸𝐷 is similar to triangle 𝐴𝐢𝐡. And in similar triangles, corresponding side lengths are proportional. We’re interested in the ratio 𝐴𝐡 over 𝐴𝐷. 𝐴𝐡 represents a larger side length, and 𝐴𝐷 is the corresponding smaller side length. This means that we are looking for the ratio that has a larger side length and the corresponding smaller side length.

Option (A) has 𝐴𝐡 corresponding to 𝐷𝐡. But 𝐷𝐡 is not part of the smaller triangle, which means option (A) cannot work. Option (B) has side length 𝐴𝐢, which is part of our larger triangle, and then the distance from 𝐴 to 𝐸. 𝐴 to 𝐸 is the corresponding smaller side length from 𝐴𝐢. This is an equal ratio. But let’s check the others just in case.

Again, we have the side length 𝐴𝐢, but the denominator is 𝐸𝐢. And 𝐸𝐢 is not part of the smaller triangle, which makes this an invalid ratio. What about option (D)? 𝐴𝐷 is a smaller side length, and then 𝐷𝐡, which is not part of our similar triangles. And we see that again 𝐴𝐸 is a smaller triangle, but 𝐸𝐢 is not part of any of our similar triangles. The only ratio that is equal to 𝐴𝐡 over 𝐴𝐷 in this list is 𝐴𝐢 over 𝐴𝐸.

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