### Video Transcript

If πΈπΆ equals 19 centimeters,
determine the length of πΈπ· rounded to the nearest hundredth, if necessary.

We can fill in the information on
the diagram that πΈπΆ equals 19 centimeters. We can also establish that the
length πΈπ· that we wish to find out is on the top part of this diagram. It might not look particularly
obvious how we would find this length πΈπ·, but there is one way we could check. If our smaller triangle πΈπΆπ΄ is
similar to the larger triangle of πΈπ·π΅, then that would give us a way to find the
missing length of πΈπ·.

Letβs have a look at both of these
triangles and jot down anything that we know about the angles. From the markings on the diagram,
we can see that angle π΄πΆπΈ in triangle πΈπΆπ΄ is equal to angle π΅π·πΈ in triangle
πΈπ·π΅. Weβre not given any information
about the angle πΆπ΄πΈ. So, letβs look at the other angle,
which is angle πΆπΈπ΄. There is, in fact, another equal
angle to this in triangle πΈπ·π΅. And itβs this one, angle
π·πΈπ΅. We know this because these two
angles are vertically opposite.

What weβve found now then is that
there are two pairs of corresponding angles congruent in our two triangles. We can then write that triangle
πΈπΆπ΄ and triangle πΈπ·π΅ are similar using the π΄π΄ similarity rule, where the
π΄π΄ stands for two pairs of corresponding angles congruent.

So, how does this help us find our
missing length of πΈπ·? Well, in similar triangles, the
corresponding sides are in proportion. So, we need to find a pair of
corresponding sides and work out the proportion. Weβre given the length of a pair of
corresponding sides, the length π·π΅ and the length π΄πΆ. So, if we were going from the
smaller triangle πΈπΆπ΄ to the larger triangle πΈπ·π΅, we could find the proportion
of sides or the scale factor by working out the new length divided by the original
length.

So, our scale factor would be our
new length, in this case, the length of π΅π·, over the length of π΄πΆ, which is
nine. We could simplify this fraction to
give us a scale factor of two, which means that every length on the smaller triangle
is multiplied by two to give the corresponding length on the larger triangle. We now need to work out which
length corresponds to the length of πΈπ· that we wish to find out. And itβs this length, πΆπΈ.

To find the length of πΈπ· then, we
take the corresponding length on triangle πΈπΆπ΄, which is πΆπ΄. And itβs 19 centimeters. And we multiply it by the scale
factor of two. And 19 multiplied by two will give
us 38 because 10 times two is 20 and nine times two is 18. Adding those would give us 38. The length units here will still be
in centimeters. So, we found the length of πΈπ· as
38 centimeters. And we did this firstly by proving
that the two triangles were similar.