### Video Transcript

If π»π· equals 17.5 centimeters, π·π΄ equals 11.3 centimeters, and πΆπ΅ equals 70 centimeters, find the length of π΄πΆ.

Letβs begin by filling in the given measurements onto the diagram. So we have π»π· is 17.5 centimeters, π·π΄ is 11.3 centimeters, and πΆπ΅ equals 70 centimeters. The length that we want to calculate is π΄πΆ. It might be useful to see if we can establish if the triangle π΄π·π» and π΄πΆπ΅ are similar. Two of the ways that we can show similarity are by using the AA rule or the SSS rule. In the AA rule, we show that there are two pairs of corresponding angles congruent. In the SSS rule, we show that there are three pairs of corresponding sides in proportion.

In this question, we can see that weβre not given enough information about the sides. So letβs see if we can use the AA rule. Weβre not given any angle measurements. But if we look at this angle, π»π΄π·, thereβs a congruent angle to it. And thatβs at the angle πΆπ΄π΅. This is because we have a pair of vertically opposite angles. Looking at the angle π΄π»π· and using the fact that we have a pair of parallel lines and a transversal π»π΅, then the angle π΄π΅πΆ would be congruent to this one. This means that weβve found two pairs of corresponding angles congruent. And itβs sufficient to say that our triangles π΄π»π· and π΄π΅πΆ are similar.

Notice that we couldβve also used the angles π»π·π΄ and π΄πΆπ΅ to show another pair of corresponding congruent angles. In a triangle, knowing that two pairs of corresponding angles are congruent automatically means that the final pair of corresponding angles are also congruent. So now that weβve shown that we have similar triangles, letβs see if we can work out the length of π΄πΆ.

In similar triangles, the sides are in proportion, so letβs see if we can work out this proportion. Weβre given the lengths of πΆπ΅ and π»π·, and these two sides are corresponding. We want to work out the length π΄πΆ, so weβll need to establish which side is corresponding to this one. Well, itβs the length π΄π·. When weβre writing our proportion relationship, we want to make sure that we get our lengths π΄πΆ and π΄π· in the correct place. π΄πΆ is part of the triangle that also includes πΆπ΅. And π΄π· is part of the triangle that includes the length π»π·.

We can now fill in the lengths that we know. πΆπ΅ is 70 centimeters, π»π· is 17.5 centimeters, and π΄π· is 11.3 centimeters. We can take the cross-product to find our missing length for π΄πΆ. This gives us π΄πΆ times 17.5 equals 70 times 11.3. Evaluating the right-hand side gives us 17.5 times π΄πΆ equals 791. Dividing both sides by 17.5 gives us that π΄πΆ equals 45.2. And the units here will be centimeters.

An alternative method of working out could have included that the scale factor from the triangle π΄π·π» to triangle π΄π΅πΆ is four. So we would multiply the lengths by four. Multiplying π΄π·, which is 11.3 centimeters, by four wouldβve given us that π΄πΆ is equal to 45.2 centimeters. Either method would lead us to the answer that π΄πΆ is 45.2 centimeters.