### Video Transcript

Given that π΄π΅ equals 24
centimeters, π΄π· equals 36 centimeters, π΄πΆ equals 18 centimeters, and πΈπ equals
15 centimeters, find the length of line segment π΄πΈ and line segment π·π.

Starting by filling in the lengths
on the diagram, we can also see that we have three parallel lines marked on this
diagram. And we also have two
transversals. A transversal is a line which
passes through two lines in the same plane at two distinct points. We can use the fact that if three
or more parallel lines are cut by two transversals, then they divide the
transversals proportionally.

As weβre given the lengths on the
line π·π΅, we can use this to work out the proportional relationship between the
segments. We can write that the segment π·π΄
over π΄π΅ is equal to the segment π΄πΈ over π΄πΆ. And we can then substitute the
values that weβre given for each segment. On the left-hand side, we have the
values of 36 over 24, which is equal to π΄πΈ over 18.

To find the missing value for π΄πΈ,
we will take the cross product. Noticing that 12 is a factor of
both 36 and 24 means that we can simplify the fraction on the left-hand side as
three over two. And thatβs equal to π΄πΈ over
18. Taking the cross product then, we
have three times 18 is equal to two times π΄πΈ. So 54 equals two times π΄πΈ. Dividing both sides by two will
give us that 27 equals π΄πΈ. And therefore, π΄πΈ is equal to 27
centimeters. And we found our first missing
length.

To find the next missing length of
π·π, we can notice that, on the other transversal, the length πΈπ will be the
corresponding length. Defining the length π·π as π, we
can then write that π over 24 is equal to 15 over 18. We can simplify the fraction 15
over 18 to give us π over 24 equals five-sixths. We can then take the cross
product. So π times six or six π is equal
to 24 times five. And simplifying, we have six π
equals 120. So π equals 20. And given that we define the length
π·π as π, then we have π·π equals 20 centimeters.

So our final answer is π΄πΈ equals
27 centimeters. π·π equals 20 centimeters.