In the figure, lines 𝐿 sub one, 𝐿
sub two, 𝐿 sub three, and 𝐿 sub four are parallel. Given that 𝑋𝑍 equals 12, 𝑍𝑁
equals eight, 𝐴𝐵 equals 10, and 𝐵𝐶 equals five, what is the length of line
Let’s start by filling in the
measurements of the line segments that we’re given. We can see in the diagram that
there are four parallel lines, 𝐿 one to 𝐿 four, and there are two transversals, 𝑀
and 𝑀 prime. A transversal is a line which
passes through two lines in the same plane at two distinct points.
To find the missing length of 𝐶𝐷,
we’re going to use a key fact about parallel lines and the proportionality of the
transversals. And that is, if three or more
parallel lines are cut by two transversals. Then, they divide the transversal
proportionally. So taking a look on our transversal
𝑀 prime, we can write that our segment 𝑍𝑁 over the segment 𝑋𝑍 has a
proportional relationship to the transversal 𝑀, which is equal to the segment 𝐶𝐷
over the segment 𝐴𝐶.
If we define our unknown length
𝐶𝐷 as 𝑥, then we can fill in the values that we have for our line segments. Since 𝑍𝑁 is eight and 𝑋𝑍 is 12,
we can write eight over 12. On the transversal 𝑀, we have our
unknown length 𝐶𝐷 as 𝑥. And the line 𝐴𝐶 is made up of the
sections five and 10, which is 15. We can simplify our fraction eight
12ths by dividing the numerator and denominator by four, which leaves us with the
equation two-thirds equals 𝑥 over 15.
To solve this, we take the cross
product. So, we have two times 15 which is
30 is equal to three 𝑥. And therefore, to find 𝑥, we
divide both sides by three, which gives us that 10 equals 𝑥, where 𝑥 is 10. And since 𝑥 corresponds to the
length of 𝐶𝐷, then we have our final answer that the length of 𝐶𝐷 is 10.