### Video Transcript

Given that ๐๐ฟ equals nine centimeters, find the length of line segment ๐๐.

Letโs begin by observing that the lines ๐ด๐, ๐ต๐, ๐ถ๐, and ๐ท๐ฟ are all parallel lines. Alongside this, we observe that we have a transversal ๐ด๐ท. Aside from the information that the line segment ๐๐ฟ is nine centimeters, the only other measurement clue weโre given is that these three line segments ๐ด๐ต, ๐ต๐ถ, and ๐ถ๐ท are congruent.

In order to find the length of line segment ๐๐, weโll need to use Thalesโs 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 Thalesโs special theorem. We might wonder if this theorem means that the line segments ๐ด๐ต and ๐๐ are congruent. In fact, it does not. It means that because line segments ๐ด๐ต, ๐ต๐ถ, and ๐ถ๐ท are congruent, then the line segments ๐๐, ๐๐, and ๐๐ฟ 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 ๐๐, remember that we were given that ๐๐ฟ is nine centimeters. Letโs write out some of the things that we know. Firstly, we know that the whole of the line segment ๐๐ฟ consists of ๐๐ plus ๐๐ plus ๐๐ฟ. But we know that each of these line segments are congruent. We could even say that ๐๐ is equal to ๐๐ and ๐๐ฟ is also equal to ๐๐. We could therefore write that ๐๐ฟ is equal to three times ๐๐.

Given the information that ๐๐ฟ is nine centimeters, we can write that nine is equal to three times ๐๐. When we divide both sides by three, we get three is equal to ๐๐. And so ๐๐ 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 wonโt of course be the final answer. We still need to work out ๐๐. Since the line segment of ๐๐ is made up of two line segments of three centimeters, then we can give the answer that the length of the line segment ๐๐ is six centimeters.