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

Given that π΄π΅πΆπ· βΌ πΈπΉπΊπ», determine the length of line segment πΊπ».

03:16

### Video Transcript

Given that π΄π΅πΆπ· is similar to πΈπΉπΊπ», determine the length of line segment πΊπ».

Here, we are told that the two quadrilaterals π΄π΅πΆπ· and πΈπΉπΊπ» 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 πΊπ».

In the figure, we are given the lengths of πΉπΊ and π΅πΆ 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 πΉπΊ over π΅πΆ. So, if we take the side that we need to calculate, thatβs πΊπ», the corresponding side in π΄π΅πΆπ· is πΆπ·. So we can say that the proportion of πΉπΊ over π΅πΆ is equal to πΊπ» over πΆπ·.

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 πΉπΊ as 96 inches over π΅πΆ, which is 32 inches, equals πΊπ» over πΆπ·, which is 35 inches. Simplifying the left-hand side, we have three equals πΊπ» over 35. Then, we can multiply both sides to give us that πΊπ» equals 105.

So, including the length units in our answer, we have determined that the length of πΊπ» 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 πΊπ».

To work out the scale factor from π΄π΅πΆπ· to πΈπΉπΊπ», we would also begin by finding a pair of corresponding sides whose length we are given. We would use the same sides of πΉπΊ and π΅πΆ 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 πΈπΉπΊπ» is three times the length of the corresponding side in π΄π΅πΆπ·. So, identifying that πΊπ» and πΆπ· are corresponding, we would simply take the length of 35 inches for πΆπ· and multiply it by three. This would give us the same answer of 105 inches for the length of πΊπ».