### 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.