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

Video Transcript

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

The first thing we can note is that the four lines 𝐴𝑋, 𝐵𝑌, 𝐶𝑍, and 𝐷𝐿 are all marked as parallel on the diagram. We also have three congruent line segments shown. So 𝐴𝐵, 𝐵𝐶, and 𝐶𝐷 are all equal in length. We are given the information that the line segment 𝑋𝐿 is nine centimeters long. And we need to determine the length of line segment 𝑋𝑍.

Now, this may seem impossible, until we recall that there is a property that we can apply here, which involves parallel lines and transversals. If a set of parallel lines divide a transversal into segments of equal length, then they divide any other transversal into segments of equal length. So, because we do have the transversal line 𝐴𝐷 divided into congruent line segments by the four parallel lines, then the same parallel lines also divide the other transversal line 𝑋𝐿 into congruent line segments such that 𝑋𝑌 equals 𝑌𝑍 equals 𝑍𝐿. And because these line segments are all congruent and sum to nine centimeters, then they all must be three centimeters each.

So, adding the lengths of line segments 𝑋𝑌 and 𝑌𝑍 of three and three centimeters each would give us the answer that the length of line segment 𝑋𝑍 is six centimeters.