Video Transcript
Given that 𝐴𝐵𝐶𝐷 is similar to
𝐸𝐹𝐺𝐻, determine the length of line segment 𝐺𝐻.
Here, we are told that the two
quadrilaterals 𝐴𝐵𝐶𝐷 and 𝐸𝐹𝐺𝐻 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 𝐺𝐻.
In the figure, we are given the
lengths of 𝐹𝐺 and 𝐵𝐶 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
𝐹𝐺 over 𝐵𝐶. So, if we take the side that we
need to calculate, that’s 𝐺𝐻, the corresponding side in 𝐴𝐵𝐶𝐷 is 𝐶𝐷. So we can say that the proportion
of 𝐹𝐺 over 𝐵𝐶 is equal to 𝐺𝐻 over 𝐶𝐷.
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 𝐹𝐺 as 96 inches over
𝐵𝐶, which is 32 inches, equals 𝐺𝐻 over 𝐶𝐷, which is 35 inches. Simplifying the left-hand side, we
have three equals 𝐺𝐻 over 35. Then, we can multiply both sides to
give us that 𝐺𝐻 equals 105.
So, including the length units in
our answer, we have determined that the length of 𝐺𝐻 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 𝐺𝐻.
To work out the scale factor from
𝐴𝐵𝐶𝐷 to 𝐸𝐹𝐺𝐻, we would also begin by finding a pair of corresponding sides
whose length we are given. We would use the same sides of 𝐹𝐺
and 𝐵𝐶 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 𝐸𝐹𝐺𝐻 is
three times the length of the corresponding side in 𝐴𝐵𝐶𝐷. So, identifying that 𝐺𝐻 and 𝐶𝐷
are corresponding, we would simply take the length of 35 inches for 𝐶𝐷 and
multiply it by three. This would give us the same answer
of 105 inches for the length of 𝐺𝐻.
