# Question Video: Finding the Side Length of a Parallelogram Using the Similarity of Triangles Mathematics

Given that π΄π΅πΆπ· is a parallelogram, π΅ is the midpoint of line segment π΄πΉ, πΆπΈ = 8 cm, π·πΈ = 16 cm, and πΆπ = 11 cm, find the length of line segment π΄π·.

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### Video Transcript

Given that π΄π΅πΆπ· is a parallelogram, π΅ is the midpoint of line segment π΄πΉ, πΆπΈ equals eight centimeters, π·πΈ equals 16 centimeters, and πΆπ equals 11 centimeters, find the length of line segment π΄π·.

In this question, weβre told firstly that π΄π΅πΆπ· is a parallelogram. Thatβs 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 πΆπ΅ and line πΆπ· is parallel to π΅π΄. The next thing weβre told is that π΅ is the midpoint of line segment π΄πΉ. And weβre given the measurements that πΆπ΄ is eight centimeters, π·π΄ is 16 centimeters, and πΆπ 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 theyβre parallel, but we should also recall that opposite sides are congruent. That means theyβre the same length. Therefore, if we need a length of the opposite side to π΄π·, the length of the line segment πΆπ΅, then we would know that it would be of the same length. The length of 11 centimeters that weβre given here only refers to the line segment πΆπ and not to the entire line segment of πΆπ΅. So letβs see if thereβs a way in which we could calculate this line segment ππ΅.

Letβs consider these two triangles. We have the smaller triangle πΆπΈπ and the larger triangle of π΅πΉπ. We might then ask ourselves if these two triangles are similar. That means theyβll 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 letβs check out the angles in these triangles. Letβs look at this angle πΈππΆ and see if there would be an equal angle to this one. Well, in fact, there would. Thereβs a vertically opposite angle, the angle π΅ππΉ. So these two angles will be equal in size.

Next, letβs look at the angle πΈπΆπ. If we consider that we have the parallel lines π·πΆ and π΄πΉ, then the transversal of πΆπ΅ would give us an equal angle. We can say that angle πΆπ΅πΉ is equal to the angle π΄πΆπ because we have alternate angles. As weβve demonstrated that there are two pairs of corresponding angles equal, then weβve shown that the AA rule is satisfied. This means that our two triangles, πΆπΈπ and π΅πΉπ, are indeed similar. So letβs return to the reason that we wanted to check these two triangles were similar, and thatβs to find the length of the line segment ππ΅.

In these similar triangles, theyβll 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, weβre not given the length of this line segment π΅πΉ, but we can in fact work it out relatively simply. Because weβre told that π΅ is the midpoint of π΄πΉ, then the length of the line segment π΅πΉ 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 thatβs 24 centimeters. And so, π΅πΉ 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 πΆπ΄ is eight centimeters and π΅πΉ 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 weβre 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 πΆπ. We can therefore take πΆπ 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, weβre about ready to calculate the length of the line segment π΄π·. Remembering that the parallelogram has opposite sides congruent, then weβre adding 11 centimeters and 33 centimeters, which gives us the final answer that the length of line segment π΄π· is 44 centimeters.