Question Video: Finding the Unknown Lengths in a Triangle given the Other Sides’ Lengths Using the Relations of Parallel Lines Mathematics

Video Transcript

In the figure, the sides 𝑋𝑌 and 𝐵𝐶 are parallel. If 𝐴𝑋 equals 18, 𝑋𝐵 equals 24, and 𝐴𝑌 equals 27, what is the length of 𝑌𝐶?

We’re given the side lengths 𝐴𝑋, 𝑋𝐵, and 𝐴𝑌. And we want to find the length of 𝑌𝐶. We’re also told that the sides 𝑋𝑌 and 𝐵𝐶 are parallel. Now, the side splitter theorem tells us that if a line parallel to one side of a triangle intersects the other two sides, then that line divides those two sides proportionally. In particular, in our case this means that 𝐴𝑌 is to 𝑌𝐶 as 𝐴𝑋 is to 𝑋𝐵. Now, if we substitute our known lengths into this equation, 𝐴𝑌 is 27, 𝐴𝑋 is 18, and 𝑋𝐵 is 24, we have 27 over 𝑌𝐶 is equal to 18 over 24. We can rearrange this to get 𝑌𝐶 equal to 24 over 18 times 27, which evaluates to 36.

And so using the side splitter theorem, we find that the length of 𝑌𝐶 is 36 units.