### Video Transcript

Given that ππΏ equals nine centimeters, find the length of line segment ππ.

Letβs begin by observing that the lines π΄π, π΅π, πΆπ, and π·πΏ are all parallel lines. Alongside this, we observe that we have a transversal π΄π·. Aside from the information that the line segment ππΏ is nine centimeters, the only other measurement clue weβre given is that these three line segments π΄π΅, π΅πΆ, and πΆπ· are congruent.

In order to find the length of line segment ππ, weβll need to use Thalesβs 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 Thalesβs special theorem. We might wonder if this theorem means that the line segments π΄π΅ and ππ are congruent. In fact, it does not. It means that because line segments π΄π΅, π΅πΆ, and πΆπ· are congruent, then the line segments ππ, ππ, and ππΏ 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 ππ, remember that we were given that ππΏ is nine centimeters. Letβs write out some of the things that we know. Firstly, we know that the whole of the line segment ππΏ consists of ππ plus ππ plus ππΏ. But we know that each of these line segments are congruent. We could even say that ππ is equal to ππ and ππΏ is also equal to ππ. We could therefore write that ππΏ is equal to three times ππ.

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