### Video Transcript

Given that πΈπ· equals 7.5
centimeters, find the lengths of line segment π΄πΆ and line segment π΅πΈ.

We can start by adding the length
of the line segment πΈπ·, which is 7.5 centimeters, to the diagram. The two lengths of π΄πΆ and π΅πΈ
that we need to calculate are marked on the diagram.

Now, it doesnβt appear as though
there is enough information to help us work out these lengths. However, the two congruent line
segments will be very helpful. We can note that the triangle
π΄πΆπ· must be a right triangle, since we have the angle measure of angle π΄π·πΆ
marked as 90 degrees. We can observe that the line
segment π·πΈ is from one of the vertices of this triangle to the midpoint of the
opposite side.

Remember, we can say that πΈ is the
midpoint of the side π΄πΆ because it bisects this line segment into two congruent
lengths. Therefore, we can describe the line
segment π·πΈ as a median of triangle π΄πΆπ·. And there is a useful theorem
regarding a median such as this. It is that in a right triangle, the
length of the median drawn from the vertex of the right angle equals half the length
of the triangleβs hypotenuse. So, in this triangle, the median
πΈπ· is half the length of π΄πΆ, which is the hypotenuse of the right triangle. Alternatively, by multiplying both
sides by two, we could write this equation as two times πΈπ· equals π΄πΆ. We know that πΈπ· is 7.5
centimeters. So two times this gives us that
π΄πΆ is 15 centimeters. And that is the first required side
length found.

Next, we need to find the length of
the line segment π΅πΈ. We can consider the triangle
π΄π΅πΆ, which is also a right triangle. Then, we have the same scenario as
before. And in this triangle, we can say
that π΅πΈ is a median, as itβs a line from a vertex to the midpoint of the opposite
side. Importantly, itβs also the median
drawn from the vertex of the right angle. So this median π΅πΈ is half the
length of the hypotenuse π΄πΆ. And weβve already calculated that
π΄πΆ is 15 centimeters. So half of this is 7.5
centimeters.

We can therefore give the answers
for the lengths of both line segments. π΄πΆ is 15 centimeters, and π΅πΈ is
7.5 centimeters.