### Video Transcript

In the figure, lines πΏ sub one, πΏ
sub two, πΏ sub three, and πΏ sub four are parallel. Given that ππ equals 12, ππ
equals eight, π΄π΅ equals 10, and π΅πΆ equals five, what is the length of line
segment πΆπ·?

Letβs start by filling in the
measurements of the line segments that weβre given. We can see in the diagram that
there are four parallel lines, πΏ one to πΏ four, and there are two transversals, π
and π 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 πΆπ·,
weβre 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
π prime, we can write that our segment ππ over the segment ππ has a
proportional relationship to the transversal π, which is equal to the segment πΆπ·
over the segment π΄πΆ.

If we define our unknown length
πΆπ· as π₯, then we can fill in the values that we have for our line segments. Since ππ is eight and ππ is 12,
we can write eight over 12. On the transversal π, we have our
unknown length πΆπ· as π₯. And the line π΄πΆ 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 π₯ over 15.

To solve this, we take the cross
product. So, we have two times 15 which is
30 is equal to three π₯. And therefore, to find π₯, we
divide both sides by three, which gives us that 10 equals π₯, where π₯ is 10. And since π₯ corresponds to the
length of πΆπ·, then we have our final answer that the length of πΆπ· is 10.