# Question Video: Finding the Lengths of Proportional Line Segments between Parallel Lines

Three Parallel Lines Theorem
Three Parallel Lines Theorem

### Video Transcript

Given that ๐๐ฟ equals nine centimeters, find the length of line segment ๐๐.

Letโs begin by observing that the lines ๐ด๐, ๐ต๐, ๐ถ๐, and ๐ท๐ฟ are all parallel lines. Alongside this, we observe that we have a transversal ๐ด๐ท. Aside from the information that the line segment ๐๐ฟ is nine centimeters, the only other measurement clue weโre given is that these three line segments ๐ด๐ต, ๐ต๐ถ, and ๐ถ๐ท are congruent.

In order to find the length of line segment ๐๐, weโll need to use Thalesโs special theorem. This theorem states that if three or more parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal. In this question, this diagram may cause some confusion with Thalesโs special theorem. We might wonder if this theorem means that the line segments ๐ด๐ต and ๐๐ are congruent. In fact, it does not. It means that because line segments ๐ด๐ต, ๐ต๐ถ, and ๐ถ๐ท are congruent, then the line segments ๐๐, ๐๐, and ๐๐ฟ are congruent. But they are congruent to each other and not the line segments on the other transversal.

In order to work out the length of line segment ๐๐, remember that we were given that ๐๐ฟ is nine centimeters. Letโs write out some of the things that we know. Firstly, we know that the whole of the line segment ๐๐ฟ consists of ๐๐ plus ๐๐ plus ๐๐ฟ. But we know that each of these line segments are congruent. We could even say that ๐๐ is equal to ๐๐ and ๐๐ฟ is also equal to ๐๐. We could therefore write that ๐๐ฟ is equal to three times ๐๐.

Given the information that ๐๐ฟ is nine centimeters, we can write that nine is equal to three times ๐๐. When we divide both sides by three, we get three is equal to ๐๐. And so ๐๐ must be three centimeters. In fact, each of these line segments must be three centimeters, which makes sense because we had a line segment of nine centimeters divided into three congruent pieces.

This wonโt of course be the final answer. We still need to work out ๐๐. Since the line segment of ๐๐ is made up of two line segments of three centimeters, then we can give the answer that the length of the line segment ๐๐ is six centimeters.

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