# Question Video: Finding the Length of the Diagonal of a Trapezoid given Its Sidesβ Lengths Mathematics

In trapezoid π΄π΅πΆπ·, sides line segment π΄π· and line segment π΅πΆ are parallel, and its diagonals intersect at π. Given that π΄π· = 66, π΅πΆ = 33, and π΄πΆ = 75, what is the length of line segment ππ΄?

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### Video Transcript

In trapezoid π΄π΅πΆπ·, sides line segment π΄π· and line segment π΅πΆ are parallel, and its diagonals intersect at π. Given that π΄π· equals 66, π΅πΆ equals 33, and π΄πΆ equals 75, what is the length of line segment ππ΄?

Letβs begin this question by having a look at the diagram. We have a trapezoid π΄π΅πΆπ·, and weβre given that there is a pair of parallel sides π΄π· and π΅πΆ. Weβre also given some lengths, which we can fill onto the diagram. π΄π· is 66, π΅πΆ is 33, and π΄πΆ is 75. Weβre asked to work out the length of line segment ππ΄, which is part of the diagonal π΄πΆ. At this point, we havenβt got enough information to work out the length of line segment ππ΄. So letβs consider if we have any similar triangles in this trapezoid. Specifically, letβs look at this triangle ππ΅πΆ and this triangle ππ·π΄. Letβs check if triangle ππ΅πΆ is similar to triangle ππ·π΄. But first, letβs 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, weβre not given enough information about the sides, so letβs 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. Letβs begin with the angle ππ΅πΆ. Because we have a pair of parallel lines π΅πΆ and π΄π·, then we actually have an angle thatβs equal to angle ππ΅πΆ. Itβs this angle at ππ·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΄πΆ, we can say that angle ππΆπ΅ must be equal to angle ππ΄π· as these angles are alternate. Since weβve found two pairs of corresponding angles congruent, then this fulfills the AA criterion and proves that triangle ππ΅πΆ is similar to triangle ππ·π΄. Of course, we couldβve also shown that angle π΅ππΆ is equal to angle π·ππ΄ 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, letβs see how we can use this to help us work out this length of the line segment ππ΄. Letβs look at these corresponding sides, π΅πΆ 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 ππ΅πΆ to the larger triangle ππ·π΄ must be two. And so the length of ππ΄ that we wish to find out must be double the length of the corresponding side, which would be side πΆπ.

But we donβt actually know the length of πΆπ. So letβs see if we can use the fact that the whole length of π΄πΆ is 75. We have worked out that the scale factor from triangle ππ΅πΆ to triangle ππ·π΄ is two. So that means that we could write the ratio of the line πΆπ to the line π΄π as the ratio of one to two. In order to divide the length π΄πΆ 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 πΆπ 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 ππ΄ is 50, and if we needed to give a unit, these would be length units.