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.