Question Video: Checking Whether Two given Triangles Are Similar or Not Mathematics • 8th Grade

Video Transcript

Are the two triangles similar?

Let’s begin by recalling that similar triangles have corresponding angles congruent and corresponding sides in proportion. If we look at the smaller triangle 𝐴𝐷𝐸 and the larger triangle 𝐴𝐵𝐶, it doesn’t seem as though these two triangles would be the same shape. However, let’s see if we could prove it mathematically, just in case the triangles are not drawn correctly. In order to show that two triangles are similar, we can remember that we would either use the AA rule to show that two pairs of angles are congruent or the SSS rule to demonstrate that there are three pairs of corresponding sides in proportion. We’re not given enough information about the angles here, so let’s see if we can use the SSS rule.

We need to check if there’s the same ratio or proportion between corresponding sides. For example, is this side 𝐴𝐸 over the side 𝐴𝐶 in the same proportion as 𝐴𝐷 over 𝐴𝐵 and the same as 𝐸𝐷 over 𝐶𝐵? We can fill in the numerical values for the lengths into our side information. But as we don’t have any dimensions for sides 𝐸𝐷 and 𝐶𝐵, we won’t be able to prove the two triangles are similar. But if the proportions of 𝐴𝐸 and 𝐴𝐶 and 𝐴𝐷 and 𝐴𝐵 are different, then we could prove that they are not similar. Let’s take a look.

The length of 𝐴𝐸 is given as 46 centimeters. But be careful as the length of 𝐴𝐶 isn’t 32.2, but rather it’s the sum of 46 and 32.2, which is 78.2. 𝐴𝐷 is 22, and 𝐴𝐵 is the sum of 22 and 24.2 centimeters, which is 46.2 centimeters. We now need to compare these two fractions to see if they’re equivalent. We can begin by removing this decimal point from the denominator. We can do this by multiplying both of our numerators and denominators by 10. Dividing the numerator and denominator by 46, then 460 over 782 simplifies to 10 over 17. Dividing the numerator and denominator of 220 over 462 by 22 gives us the fraction 10 over 21. We can, therefore, see that these two ratios are not equal. Therefore, the sides are not in proportion. If the sides are not in proportion, then the triangles would not be similar. So, our answer here would be no.

Before we finish with this question, just a point to note. If we had find that 𝐴𝐸 over 𝐴𝐶 was equal to 𝐴𝐷 over 𝐴𝐵, then we would have also needed the values for 𝐸𝐷 and 𝐶𝐵. As it’s not enough just to show that two sides are in proportion, we would need to show that there are three pairs of corresponding sides in proportion. In this question, it didn’t matter that we didn’t have these two other lengths as it was enough to show that these triangles were not similar.