### Video Transcript

Given that π΄π΅πΆπ· is similar to
πΈπΉπΊπ», determine the length of line segment πΊπ».

Here, we are told that the two
quadrilaterals π΄π΅πΆπ· and πΈπΉπΊπ» are similar. We can recall that two polygons are
similar if their corresponding angles are congruent and their corresponding sides
are in proportion. We can use this proportionality to
help us work out the side length of πΊπ».

In the figure, we are given the
lengths of πΉπΊ and π΅πΆ as 96 inches and 32 inches, respectively. These two sides appear to be
corresponding sides. And we can confirm that they are by
using the similarity statement. Because these shapes are similar,
we can therefore say that all the sides will be in the same proportion as that of
πΉπΊ over π΅πΆ. So, if we take the side that we
need to calculate, thatβs πΊπ», the corresponding side in π΄π΅πΆπ· is πΆπ·. So we can say that the proportion
of πΉπΊ over π΅πΆ is equal to πΊπ» over πΆπ·.

Notice that when we are writing a
proportionality statement like this, we must be careful to keep the sides of each
shape either as both on the numerator or as both on the denominator. Now we can fill in the lengths that
we know. We have πΉπΊ as 96 inches over
π΅πΆ, which is 32 inches, equals πΊπ» over πΆπ·, which is 35 inches. Simplifying the left-hand side, we
have three equals πΊπ» over 35. Then, we can multiply both sides to
give us that πΊπ» equals 105.

So, including the length units in
our answer, we have determined that the length of πΊπ» is 105 inches. As an alternative method, we could
have calculated the scale factor between these two quadrilaterals and used that to
determine the length of πΊπ».

To work out the scale factor from
π΄π΅πΆπ· to πΈπΉπΊπ», we would also begin by finding a pair of corresponding sides
whose length we are given. We would use the same sides of πΉπΊ
and π΅πΆ as in the previous method. Writing these as the fraction 96
over 32, we could simplify this as three over one, or simply three. So every length on πΈπΉπΊπ» is
three times the length of the corresponding side in π΄π΅πΆπ·. So, identifying that πΊπ» and πΆπ·
are corresponding, we would simply take the length of 35 inches for πΆπ· and
multiply it by three. This would give us the same answer
of 105 inches for the length of πΊπ».