### Video Transcript

Given that π΄π΅πΆπ· is a parallelogram, π΅ is the midpoint of line segment π΄πΉ, πΆπΈ
equals eight centimeters, π·πΈ equals 16 centimeters, and πΆπ equals 11
centimeters, find the length of line segment π΄π·.

In this question, weβre told firstly that π΄π΅πΆπ· is a parallelogram. Thatβs the quadrilateral at the top of this figure. One of the main properties that we should remember about parallelograms are that
opposite sides are parallel. In the figure, this would mean that line π·π΄ is parallel to line πΆπ΅ and line πΆπ·
is parallel to π΅π΄. The next thing weβre told is that π΅ is the midpoint of line segment π΄πΉ. And weβre given the measurements that πΆπ΄ is eight centimeters, π·π΄ is 16
centimeters, and πΆπ equals 11 centimeters, so we can fill in these measurements
onto the diagram.

The question asks us to find the length of this line segment π΄π·. In order to work this out, we should recall another important fact about the opposite
sides in a parallelogram. We know that theyβre parallel, but we should also recall that opposite sides are
congruent. That means theyβre the same length. Therefore, if we need a length of the opposite side to π΄π·, the length of the line
segment πΆπ΅, then we would know that it would be of the same length. The length of 11 centimeters that weβre given here only refers to the line segment
πΆπ and not to the entire line segment of πΆπ΅. So letβs see if thereβs a way in which we could calculate this line segment ππ΅.

Letβs consider these two triangles. We have the smaller triangle πΆπΈπ and the larger triangle of π΅πΉπ. We might then ask ourselves if these two triangles are similar. That means theyβll be the same shape but a different size. One way that we can prove similarity in triangles is by using the AA criterion in
which we see if there are two pairs of corresponding pairs of angles congruent. So letβs check out the angles in these triangles. Letβs look at this angle πΈππΆ and see if there would be an equal angle to this
one. Well, in fact, there would. Thereβs a vertically opposite angle, the angle π΅ππΉ. So these two angles will be equal in size.

Next, letβs look at the angle πΈπΆπ. If we consider that we have the parallel lines π·πΆ and π΄πΉ, then the transversal of
πΆπ΅ would give us an equal angle. We can say that angle πΆπ΅πΉ is equal to the angle π΄πΆπ because we have alternate
angles. As weβve demonstrated that there are two pairs of corresponding angles equal, then
weβve shown that the AA rule is satisfied. This means that our two triangles, πΆπΈπ and π΅πΉπ, are indeed similar. So letβs return to the reason that we wanted to check these two triangles were
similar, and thatβs to find the length of the line segment ππ΅.

In these similar triangles, theyβll be the same proportion between every length on
the smaller triangle to every length on the larger triangle. We could also think of this in terms of finding the scale factor. In order to find this scale factor, we need a corresponding pair of lengths on the
smaller triangle and on the larger triangle. Now, weβre not given the length of this line segment π΅πΉ, but we can in fact work it
out relatively simply. Because weβre told that π΅ is the midpoint of π΄πΉ, then the length of the line
segment π΅πΉ will be the same as the length of this line segment π΄π΅.

Using the fact that, in a parallelogram, opposite sides are parallel and congruent,
then the length of π΄π΅ will be equal to the sum of eight centimeters and 16
centimeters. And thatβs 24 centimeters. And so, π΅πΉ is also 24 centimeters. We now have a pair of corresponding sides in the smaller triangle and the larger
triangle, which would allow us to work out the scale factor. If πΆπ΄ is eight centimeters and π΅πΉ is 24 centimeters, then the scale factor must
be three. That means each length on the smaller triangle could be multiplied by three to give
the corresponding length on the larger triangle.

We remember that weβre trying to find this length of the line segment π΅π, so which
side on the smaller triangle would be corresponding with this length? It would be this one, the line segment πΆπ. We can therefore take πΆπ of 11 centimeters and multiply it by the scale factor of
three to give us a length of 33 centimeters. And finally, after all of that working out, weβre about ready to calculate the length
of the line segment π΄π·. Remembering that the parallelogram has opposite sides congruent, then weβre adding 11
centimeters and 33 centimeters, which gives us the final answer that the length of
line segment π΄π· is 44 centimeters.