Video: Finding a Side Length in a Triangle given the Corresponding Side in a Similar Triangle and the Ratio of Similarity between Them

Video Transcript

If 𝐶𝐸 equals 𝑥 plus two centimeters, what is 𝑥?

First of all, on our figure, we can label 𝐶𝐸 as 𝑥 plus two centimeters. And then we should think about what else we know based on the figure. First of all, we see that line segment 𝐸𝐷 is parallel to line segment 𝐶𝐵. And then we can say that line segment 𝐴𝐵 and line segment 𝐴𝐶 are transversals of these two parallel lines. Based on these two facts, we can draw some conclusions. We can say that the parallel lines 𝐸𝐷 and 𝐶𝐵 cut this triangle proportionally. So we can say line segment 𝐴𝐸 over line segment 𝐴𝐷 will be equal to line segment 𝐶𝐸 over line segment 𝐷𝐵 by parallel lines and transversal properties.

To solve then, we can just plug in the values that we know for these line segments. Six over 𝑥 plus two is equal to four over eight. The first way we could solve this is by using cross multiplication. We can say six times eight is equal to four times 𝑥 plus two. Therefore, 48 equals four times 𝑥 plus two. And if we divide both sides of the equation by four, we see that 12 is equal to 𝑥 plus two. So we subtract two from both sides, and we see that 𝑥 equals 10. Now, I said this is one way to solve. And that’s because if we think about proportionality, and we know that the parallel lines cut these line segments proportionally, we notice that line segment 𝐷𝐵 is two times line segment 𝐴𝐷.

And in order for things to be proportional, that would mean that the same thing would have to be true on the other side. This means that 𝑥 plus two must be equal to six times two, which again shows us that side length 𝐶𝐸 must be equal to 12 and therefore 𝑥 plus two must be equal to 12. So again, 𝑥 equals 10.