Question Video: Using Properties of Similar Triangles to Calculate Lengths of Corresponding Sides Mathematics • First Year of Secondary School

Video Transcript

Triangles 𝐴𝐵𝐶 and 𝐴 prime 𝐵 prime 𝐶 prime are similar. Work out the value of 𝑥. Work out the value of 𝑦.

In this question, we have two triangles which we’re told are similar. We should remember that when we have similar triangles, this means that corresponding pairs of angles are congruent or equal and corresponding pairs of sides are in the same proportion. We can use this fact to help us work out the values of 𝑥 and 𝑦. So let’s start with the first question to find the value of 𝑥. When we’re working with similar shapes, we’re usually given or can easily calculate the length of two corresponding sides. In this diagram, we’re given the lengths of two corresponding sides 𝐴𝐵 and 𝐴 prime 𝐵 prime. These lengths are five and three.

We could write this proportion as 𝐴𝐵 over 𝐴 prime 𝐵 prime. As we need to find the length of 𝑥 which is on the line segment 𝐴 prime 𝐶 prime, then the corresponding side will be the line segment 𝐴𝐶. So the proportion of 𝐴𝐵 over 𝐴 prime 𝐵 prime must be equal to the proportion of 𝐴𝐶 over 𝐴 prime 𝐶 prime. Notice that we could’ve written this proportionality statement with the numerators and denominators both switched. But now all we need to do is fill in the values of the lengths that we’re given. So we have five over three is equal to 10 over 𝑥.

To solve for 𝑥, we can begin by taking the cross product. And this gives us that five 𝑥 is equal to 30. And then dividing both sides by five, we would get that 𝑥 must be equal to six. And so that’s the answer for the first part of the question.

Let’s now look at the second part of this question to find the value of 𝑦. This triangle will still have the ratio of sides, the same as the proportion 𝐴𝐵 over 𝐴 prime 𝐵 prime. But this time, let’s look at the other two corresponding sides. These sides are 𝐵𝐶 and 𝐵 prime 𝐶 prime. Because we have side 𝐴𝐵 on the numerator, then side 𝐵𝐶 must also be on the numerator, as it’s part of the same triangle.

So 𝐴𝐵 over 𝐴 prime 𝐵 prime must be equal to 𝐵𝐶 over 𝐵 prime 𝐶 prime. When we fill in the values, we’ll have five-thirds on the left-hand side. 𝐵𝐶 is equal to seven, and 𝐵 prime 𝐶 prime is equal to 𝑦. When we take the cross product at this time, we’ll have five times 𝑦, which is five 𝑦, is equal to three times seven, which is 21. Dividing both sides of this equation by five gives us that 𝑦 is equal to 21 over five.

We’ve now fully answered the question. 𝑥 is equal to six, and 𝑦 is equal to 21 over five. Both of these values are lengths, so the units of these would be length units.