Question Video: Finding the Side Length of a Triangle Using the Similarity of Triangles Mathematics

Video Transcript

Given that 𝐴𝐵 equals 13 centimeters, 𝐵𝐶 equals eight centimeters, 𝐴𝐶 equals 11.36 centimeters, and 𝐴𝐷 equals 10 centimeters, determine the length of 𝐴𝐸 approximated to the nearest hundredth.

In this diagram, there’s a larger triangle 𝐴𝐵𝐶 and a smaller triangle 𝐴𝐷𝐸. In order to work out the missing length, it might be sensible to work out if these triangles are similar. Similar triangles have corresponding angles congruent and corresponding sides in proportion.

Let’s fill in the lengths that we were given onto our diagram. 𝐴𝐵 is 13 centimeters, 𝐴𝐶 is 11.36 centimeters, and 𝐴𝐷 equals 10 centimeters. In order to prove that two triangles are similar, we can use the AA rule, where we show that two pairs of angles are congruent. The SSS rule, where we show that we have three pairs of sides in proportion. Or the SAS rule, where we show that we have two pairs of sides in proportion and the included angle is congruent.

In the diagram, it doesn’t look as though we have enough information for the sides to use the SSS rule, so let’s have a look at the angles. If we begin by looking at this angle 𝐸𝐴𝐷 in our smaller triangle, it would be a common angle to the angle 𝐶𝐴𝐵 in the larger triangle. So, we can say that this pair of angles is congruent. The angle marked in green, angle 𝐷𝐸𝐴 in the smaller triangle, is congruent with angle 𝐵𝐶𝐴 in the larger triangle because we have corresponding angles from the parallel lines and the transversal.

In the same way, angle 𝐴𝐷𝐸 is corresponding with angle 𝐴𝐵𝐶 in the larger triangle. So, we have another pair of congruent angles. So, now, we’ve found that there are three pairs of corresponding angles congruent. We only would have needed to show two pairs of these in order to prove that these two triangles are similar. So, now, we know that we have two similar triangles, let’s work out our missing length 𝐴𝐸.

In order to do this, we need to find the scale factor between 𝐴𝐵𝐶 and 𝐴𝐷𝐸. Often, it’s helpful to draw the triangles separately in order to help us work out the scale factor. In order to work out the scale factor from the larger triangle to the smaller triangle, we can use the rule that the scale factor is equal to the new length over the original length. We’re given the lengths of a pair of corresponding sides, 10 centimeters for 𝐴𝐷 and 13 centimeters for 𝐴𝐵.

If we’re taking the direction of the scale factor to be going towards the smaller triangle, then the new length would be 10 centimeters and the original length would be 13 centimeters. So, our scale factor will be 10 over 13. In order to work out the length of 𝐴𝐸, we take the corresponding side 𝐴𝐶 of 11.36 and multiply it by the scale factor of 10 over 13. As we’re asked for an answer to the nearest hundredth, we can reasonably use a calculator here to work out the value.

Therefore, 𝐴𝐸 is equal to 8.73846 and so on centimeters. Rounding to the nearest hundredth means that we check our third decimal digit to see if it’s five or more. As it is, then our answer rounds up to 8.74 centimeters. So, we found this value of 𝐴𝐸 by proving that the triangles were similar and working out the scale factor.