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Question Video: Finding the Unknown Lengths in a Triangle given the Other Sidesโ€™ Lengths Using the Relations of Parallel Lines Mathematics • 11th Grade

๐น๐ท๐ธ๐ถ is a parallelogram, where ๐น and ๐ท are the midpoints of line segment ๐ด๐ต and line segment ๐ด๐ถ, respectively, and ๐ถ๐ธ = 6 cm. Determine the length of ๐ต๐ถ.

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Video Transcript

๐น๐ท๐ธ๐ถ is a parallelogram, where ๐น and ๐ท are the midpoints of line segment ๐ด๐ต and line segment ๐ด๐ถ, respectively, and ๐ถ๐ธ equals six centimeters. Determine the length of ๐ต๐ถ.

Letโ€™s take the information weโ€™re given and add that in to our diagram. If ๐น๐ท๐ธ๐ถ is a parallelogram, the opposite sides are parallel, meaning ๐น๐ท is parallel to ๐ถ๐ธ and ๐น๐ถ is parallel to ๐ท๐ธ. We also know that ๐ถ๐ธ measures six centimeters. But based on our parallelogram properties, we also know that opposite side lengths will be equal. And that means that ๐ท๐น must also measure six centimeters.

But our missing side length is ๐ต๐ถ. And so weโ€™ll need to be able to say something else here. If ๐ถ๐ธ is parallel to ๐น๐ท, we can also say that ๐ต๐ธ is parallel to ๐น๐ท. And if we can say that line segment ๐ต๐ถ is parallel to line segment ๐น๐ท, then we have a line that is parallel to a side of our triangle. And that line intersects the other two sides, which means the line segment ๐น๐ท creates two similar triangles. The smaller triangle, triangle ๐ด๐ท๐น, is similar to the larger triangle, triangle ๐ด๐ถ๐ต. And in similar triangles, corresponding side lengths are proportional. The side length ๐ด๐ท on the smaller triangle corresponds to the side length ๐ด๐ถ on the larger triangle. And that will have to be equal to the smaller triangleโ€™s line segment ๐น๐ท over the larger triangleโ€™s line segment ๐ต๐ถ.

If we try to plug in the information we know, we only end up filling in the side length ๐น๐ท, which is six centimeters. But we can think carefully about this midpoint ๐ท. The midpoint ๐ท divides line segment ๐ด๐ถ in half. And so we can say that ๐ด๐ถ is equal to two times ๐ด๐ท. What weโ€™re saying is the ratio of the smaller triangle to the larger triangle would be one-half since the larger triangle is always two times greater than the smaller triangle. And if the side lengths of the larger triangle is two times that of the smaller triangle and ๐น๐ท is equal to six centimeters, we know that ๐ต๐ถ will have to be equal to 12 centimeters, as 12 is twice six.

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