Question Video: Identifying the Equal Proportions between Corresponding Sides in Two Similar Triangles Mathematics • 11th Grade

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

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Video Transcript

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

We see from the diagram that the base of triangle 𝐴𝐸𝐷, thatβ€²s side 𝐸𝐷, is parallel to the base of triangle 𝐴𝐡𝐢, thatβ€²s side 𝐢𝐡. Since corresponding angles must be equal if the two lines are parallel, angle 𝐷𝐸𝐴 is equal to angle 𝐡𝐢𝐴, and angle 𝐸𝐷𝐴 is equal to angle 𝐢𝐡𝐴. Then side 𝐸𝐷 creates triangle 𝐴𝐷𝐸, which is similar to the larger triangle 𝐴𝐡𝐢. Since these triangles are similar, the ratios of their corresponding side lengths must be equal. In particular, 𝐴𝐸 over 𝐴𝐢 is equal to 𝐴𝐷 over 𝐴𝐡.

Now we want to find which of the given fractions is equal to 𝐴𝐡 over 𝐴𝐷. And we can do this by finding the reciprocal of both sides of our equation, which gives us 𝐴𝐢 over 𝐴𝐸 is equal to 𝐴𝐡 over 𝐴𝐷. Hence, 𝐴𝐡 over 𝐴𝐷 is equal to 𝐴𝐢 over 𝐴𝐸, which corresponds to the given option (D).

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