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Question Video: Finding the Lengths of Proportional Line Segments between Parallel Lines Mathematics • 11th Grade

Find the lengths of line segment ๐ธ๐ถ and line segment ๐ท๐ต.

02:08

Video Transcript

Find the lengths of line segment ๐ธ๐ถ and line segment ๐ท๐ต.

If we look at this figure, we see that we have a larger quadrilateral ๐ด๐ต๐‘Œ๐‘‹, and the larger quadrilateral is cut by the line segments ๐ธ๐น and ๐ถ๐ท. In addition to that, we see that we have four parallel line segments, two on the outsides of the quadrilateral and the two lines that cut the inside of the quadrilateral. And what we know is if three or more parallel lines intersect two transversals, then they cut off the transversals proportionally. This means that the transversal ๐ด๐‘‹ and the transversal ๐ต๐‘Œ are being cut proportionally by these four parallel lines.

And so we can say that ๐‘‹๐ธ over ๐‘Œ๐น will be proportional to ๐ธ๐ถ over ๐น๐ท, which means we can say 14 over eight is going to be equal to the length of ๐ธ๐ถ over 12. To solve for the length of ๐ธ๐ถ, we can do cross multiplication. 12 times 14 equals eight times ๐ธ๐ถ. 168 equals eight times ๐ธ๐ถ. To find ๐ธ๐ถ, we divide both sides of the equation by eight, and we get that ๐ธ๐ถ is equal to 21. So we can say that the measure of line segment ๐ธ๐ถ is 21 centimeters.

Now, we could set up another proportion exactly like this to find the measure of line segment ๐ท๐ต. However, because we know that these values are proportional, we should notice something else. 14 centimeters times two equals 28 centimeters. And this means that line segment ๐ท๐ต will be equal to eight centimeters times two. And so we can say that the line segment ๐ท๐ต is then equal to 16 centimeters. Weโ€™ve seen two different methods for solving proportional side lengths, and we found that ๐ธ๐ถ equals 21 centimeters and ๐ท๐ต equals 16 centimeters.

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