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.