Question Video: Finding the Length of a Side in a Quadrilateral given the Corresponding Sides in a Similar Quadrilateral and Their Lengths Mathematics

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.