### Video Transcript

In trapezoid π΄π΅πΆπ·, sides line segment π΄π· and line segment π΅πΆ are parallel, and its diagonals intersect at π. Given that π΄π· equals 66, π΅πΆ equals 33, and π΄πΆ equals 75, what is the length of line segment ππ΄?

Letβs begin this question by having a look at the diagram. We have a trapezoid π΄π΅πΆπ·, and weβre given that there is a pair of parallel sides π΄π· and π΅πΆ. Weβre also given some lengths, which we can fill onto the diagram. π΄π· is 66, π΅πΆ is 33, and π΄πΆ is 75. Weβre asked to work out the length of line segment ππ΄, which is part of the diagonal π΄πΆ. At this point, we havenβt got enough information to work out the length of line segment ππ΄. So letβs consider if we have any similar triangles in this trapezoid. Specifically, letβs look at this triangle ππ΅πΆ and this triangle ππ·π΄. Letβs check if triangle ππ΅πΆ is similar to triangle ππ·π΄. But first, letβs recall the definition for similar triangles.

Similar triangles have corresponding pairs of angles congruent and corresponding pairs of sides in proportion. One way that we could prove that triangles are similar is by using the SSS rule, which checks if three pairs of corresponding sides have the same proportion. However, weβre not given enough information about the sides, so letβs see if we can apply the AA similarity criterion. For this, we would need to check that there are two pairs of corresponding angles congruent. Letβs begin with the angle ππ΅πΆ. Because we have a pair of parallel lines π΅πΆ and π΄π·, then we actually have an angle thatβs equal to angle ππ΅πΆ. Itβs this angle at ππ·π΄. Because of the parallel lines and the transversal π΅π·, then these two angles are alternate.

In the same way, if we use our two parallel lines and the transversal this time of π΄πΆ, we can say that angle ππΆπ΅ must be equal to angle ππ΄π· as these angles are alternate. Since weβve found two pairs of corresponding angles congruent, then this fulfills the AA criterion and proves that triangle ππ΅πΆ is similar to triangle ππ·π΄. Of course, we couldβve also shown that angle π΅ππΆ is equal to angle π·ππ΄ as we have a pair of opposite angles. Any two of these three angle pairs would demonstrate similarity.

Now that we know that these two triangles are similar, letβs see how we can use this to help us work out this length of the line segment ππ΄. Letβs look at these corresponding sides, π΅πΆ and π΄π·. Because these two lines are in proportion, then that means we could actually work out the scale factor of the smaller triangle to the larger triangle. 66 is double 33, so that means that the scale factor from the smaller triangle ππ΅πΆ to the larger triangle ππ·π΄ must be two. And so the length of ππ΄ that we wish to find out must be double the length of the corresponding side, which would be side πΆπ.

But we donβt actually know the length of πΆπ. So letβs see if we can use the fact that the whole length of π΄πΆ is 75. We have worked out that the scale factor from triangle ππ΅πΆ to triangle ππ·π΄ is two. So that means that we could write the ratio of the line πΆπ to the line π΄π as the ratio of one to two. In order to divide the length π΄πΆ of 75 in the ratio one to two, we would start by dividing 75 by three, which would give us 25.

Therefore, we know that the one part of πΆπ must be equal to 25 and the two part of π΄π must be equal to two times 25 which is 50. We can give the answer then that the length of line segment ππ΄ is 50, and if we needed to give a unit, these would be length units.