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

Video Transcript

Given that 𝑋𝐿 equals nine centimeters, find the length of line segment 𝑋𝑍.

Let’s begin by observing that the lines 𝐴𝑋, 𝐵𝑌, 𝐶𝑍, and 𝐷𝐿 are all parallel lines. Alongside this, we observe that we have a transversal 𝐴𝐷. Aside from the information that the line segment 𝑋𝐿 is nine centimeters, the only other measurement clue we’re given is that these three line segments 𝐴𝐵, 𝐵𝐶, and 𝐶𝐷 are congruent.

In order to find the length of line segment 𝑋𝑍, we’ll need to use Thales’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 Thales’s special theorem. We might wonder if this theorem means that the line segments 𝐴𝐵 and 𝑋𝑌 are congruent. In fact, it does not. It means that because line segments 𝐴𝐵, 𝐵𝐶, and 𝐶𝐷 are congruent, then the line segments 𝑋𝑌, 𝑌𝑍, and 𝑍𝐿 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 𝑋𝑍, remember that we were given that 𝑋𝐿 is nine centimeters. Let’s write out some of the things that we know. Firstly, we know that the whole of the line segment 𝑋𝐿 consists of 𝑋𝑌 plus 𝑌𝑍 plus 𝑍𝐿. But we know that each of these line segments are congruent. We could even say that 𝑌𝑍 is equal to 𝑋𝑌 and 𝑍𝐿 is also equal to 𝑋𝑌. We could therefore write that 𝑋𝐿 is equal to three times 𝑋𝑌.

Given the information that 𝑋𝐿 is nine centimeters, we can write that nine is equal to three times 𝑋𝑌. When we divide both sides by three, we get three is equal to 𝑋𝑌. And so 𝑋𝑌 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 won’t of course be the final answer. We still need to work out 𝑋𝑍. Since the line segment of 𝑋𝑍 is made up of two line segments of three centimeters, then we can give the answer that the length of the line segment 𝑋𝑍 is six centimeters.