### Video Transcript

Given that π΄π΅ equals 13 centimeters, π΅πΆ equals eight centimeters, π΄πΆ equals 11.36 centimeters, and π΄π· equals 10 centimeters, determine the length of π΄πΈ approximated to the nearest hundredth.

In this diagram, thereβs a larger triangle π΄π΅πΆ and a smaller triangle π΄π·πΈ. In order to work out the missing length, it might be sensible to work out if these triangles are similar. Similar triangles have corresponding angles congruent and corresponding sides in proportion.

Letβs fill in the lengths that we were given onto our diagram. π΄π΅ is 13 centimeters, π΄πΆ is 11.36 centimeters, and π΄π· equals 10 centimeters. In order to prove that two triangles are similar, we can use the AA rule, where we show that two pairs of angles are congruent. The SSS rule, where we show that we have three pairs of sides in proportion. Or the SAS rule, where we show that we have two pairs of sides in proportion and the included angle is congruent.

In the diagram, it doesnβt look as though we have enough information for the sides to use the SSS rule, so letβs have a look at the angles. If we begin by looking at this angle πΈπ΄π· in our smaller triangle, it would be a common angle to the angle πΆπ΄π΅ in the larger triangle. So, we can say that this pair of angles is congruent. The angle marked in green, angle π·πΈπ΄ in the smaller triangle, is congruent with angle π΅πΆπ΄ in the larger triangle because we have corresponding angles from the parallel lines and the transversal.

In the same way, angle π΄π·πΈ is corresponding with angle π΄π΅πΆ in the larger triangle. So, we have another pair of congruent angles. So, now, weβve found that there are three pairs of corresponding angles congruent. We only would have needed to show two pairs of these in order to prove that these two triangles are similar. So, now, we know that we have two similar triangles, letβs work out our missing length π΄πΈ.

In order to do this, we need to find the scale factor between π΄π΅πΆ and π΄π·πΈ. Often, itβs helpful to draw the triangles separately in order to help us work out the scale factor. In order to work out the scale factor from the larger triangle to the smaller triangle, we can use the rule that the scale factor is equal to the new length over the original length. Weβre given the lengths of a pair of corresponding sides, 10 centimeters for π΄π· and 13 centimeters for π΄π΅.

If weβre taking the direction of the scale factor to be going towards the smaller triangle, then the new length would be 10 centimeters and the original length would be 13 centimeters. So, our scale factor will be 10 over 13. In order to work out the length of π΄πΈ, we take the corresponding side π΄πΆ of 11.36 and multiply it by the scale factor of 10 over 13. As weβre asked for an answer to the nearest hundredth, we can reasonably use a calculator here to work out the value.

Therefore, π΄πΈ is equal to 8.73846 and so on centimeters. Rounding to the nearest hundredth means that we check our third decimal digit to see if itβs five or more. As it is, then our answer rounds up to 8.74 centimeters. So, we found this value of π΄πΈ by proving that the triangles were similar and working out the scale factor.