Question Video: Using Proportions in a Triangle to Calculate an Unknown Length Mathematics

Find the value of π‘₯.

01:41

Video Transcript

Find the value of π‘₯.

When we look at our diagram, we see a larger triangle that is cut by the segment 𝐷𝐸. And this line segment 𝐷𝐸 is parallel to one of the side lengths 𝐡𝐢. And when a triangle is cut by a line segment that’s parallel to one of its side lengths, two similar triangles are created. So we can say that if we’re given line segment 𝐷𝐸 is parallel to 𝐡𝐢, the smaller triangle, triangle 𝐴𝐷𝐸, is similar to the larger triangle, triangle 𝐴𝐡𝐢. So we can say that 𝐴𝐷 over 𝐴𝐡 is equal to 𝐷𝐸 over 𝐡𝐢. This is because in similar triangles corresponding side lengths are proportional.

Side length 𝐴𝐷 measures 10, but what about 𝐴𝐡? And this is where we need to be very careful. Side length 𝐴𝐡 is not equal to 11. Side length 𝐴𝐡 is the full distance from vertex 𝐴 to vertex 𝐡, which is 21. So 𝐴𝐷 over 𝐴𝐡 will be equal to 10 over 21. We know that side length 𝐷𝐸 is equal to 10 as well. So we now have a proportion that says 10 over 21 is equal to 10 over 𝐡𝐢. And that means 𝐡𝐢 must be equal to 21 so that these side lengths stay in proportion. Since side length 𝐡𝐢 equals 21, our missing π‘₯-value is 21.

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