Video Transcript
In this video, we will learn how to use the properties of similar triangles to solve problems. Let’s start by recalling what it means when two triangles are similar.
We say that two triangles are similar if their corresponding angles are congruent and their corresponding sides are in proportion. So, in other words, all the corresponding angles are congruent, that’s the same size, and the sides are in proportion. If the corresponding pairs of sides were also congruent, then the triangles would be congruent themselves.
So, similar triangles will have the same shape but be a different size. Corresponding sides are in proportion, but the corresponding angles are congruent. There are a few ways in which we could prove that two triangles are similar. Let’s have a look at these next.
The first method is by using the AA rule, which is when we show that two pairs of angles are congruent. The triangles drawn here would be similar as there’s two pairs of congruent angles. So, why do we not need to show that there are three pairs of congruent angles? Well, because the sum of the angles in a triangle is fixed at 180 degrees. If we’ve shown that we have two pairs of congruent angles, then the third pair of angles in each triangle would also be congruent. We could, of course, show that there are three pairs of congruent angles, but we don’t need to. We just need to show two pairs.
The second method of proving similarity in triangles is by using the SSS rule. If we can show that we have three pairs of corresponding sides in proportion, then this would show that the triangles are similar. We do have to be careful here. If we recall that when we’re using the SSS rule to prove triangles are congruent, in that case, we would be showing that the pairs of sides are congruent. When we’re proving similarity, the SSS rule means that we’re showing that the sides are in proportion.
In the triangles below, we can see that they’re similar because every length on the smaller triangle is in proportion to those in the larger triangle. In fact, each length on the smaller triangle is half of that in the larger triangle.
When we’re discussing similar triangles, we’ll also use the words scale factor. This is the ratio of corresponding lengths in similar figures. The scale factor is always given as a multiplier. For example, we might informally say that to go from the larger triangle to the smaller triangle, we divide the lengths by two. However, to say this in a more mathematically correct way, we would say that the scale factor is one-half.
The final way we could prove similarity in triangles is by using the SAS rule. This means that we’re demonstrating that we have two pairs of sides in proportion and the included pair of angles between these two sides are congruent. Note that when we’re using the SAS rule to prove congruence, the sides and the angles would all need to be congruent. As we saw in the previous method, the sides here have to be in proportion.
So, in the triangles below, if we could show that we have a corresponding pair of sides in proportion, an included pair of angles congruent, and another pair of sides in the same proportion, then we would show that these two triangles are similar. We’ll now look at a question example where we need to find the scale factor.
In the figure, given that the two triangles are similar, what is the scale factor that would take you from the larger triangle to the smaller triangle?
We can recall that the word “similar” means the same shape but a different size. More specifically, we can say that corresponding angles are congruent and corresponding sides are in proportion. In order to find the scale factor that takes us from the large triangle to the small triangle, we need to work out the proportion of the sides. This ratio or proportion is the scale factor.
We can start by looking at the base lengths and using the helpful formula that the scale factor is equal to the new length over the original length. Then, as the new length in the smaller triangle is six and the original base length is 12 in the larger triangle, we have a scale factor of six over 12, which simplifies to one-half. So, each of the lengths in the smaller triangle will be half of the lengths in the larger triangle.
We can check our answer using the other pair of given sides. If we take the length of 11 and multiply it by the scale factor of a half, we would get 11 over two, which simplifies to five and a half or 5.5. The corresponding length on the smaller triangle was indeed 5.5. And so, we’ve confirmed our answer that the scale factor from the larger triangle to the smaller triangle is a half.
We must always make sure that the scale factor is a multiplier. So, for example, we could’ve divided the lengths on the larger triangle by two to get to the smaller triangle. But our scale factor could not be “dividing by two.” It would have to be “multiplying by a half.”
We’ll now look at some questions where we find missing lengths in similar triangles.
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.
Let’s have a look at another question.
Triangles 𝐴𝐵𝐶 and 𝐴 prime 𝐵 prime 𝐶 prime are similar. Work out the measure of angle 𝑥. Work out the value of 𝑦. Work out the value of 𝑧.
In this question, we’re told that our two triangles are similar, which means that corresponding angles are congruent and corresponding sides are in proportion. If we look at this angle at 𝐴 prime denoted by the 𝑥, then we need to work out which angle in triangle 𝐴𝐵𝐶 is corresponding to this one. Sometimes in diagrams, this isn’t always clear, but we can use the order of the letters to help us.
The angle at 𝐴 prime will correspond with the angle at 𝐴. We can write this more formally as the angle 𝐶 prime 𝐴 prime 𝐵 prime is corresponding to the angle at 𝐶𝐴𝐵, this one in pink. Both of these angles are equal, and they’re 74.5 degrees. So our answer for the first part of this question is that angle 𝑥 is 74.5 degrees. It can be tempting to think that because the triangle is larger, that the angle must also be larger. But remember that the sum of the angles in a triangle is always 180 degrees.
In the second part of this question, we’re asked to find the length of 𝑦. We’ll need to work out the proportion of the lengths or the scale factor that takes us from the smaller triangle to the larger triangle. We can use a given pair of corresponding sides. Here, we have the length 𝐴 prime 𝐵 prime is five and the length 𝐴𝐵 is two. To work out the scale factor from the smaller triangle to the larger triangle, we’ll take our new length of five and divide it by the original length of two. Therefore, if we want to find a length on the longer triangle, we take the corresponding length on the smaller triangle and multiply it by five over two.
So, the length 𝑦, which we want to calculate on the larger triangle on the line 𝐵 prime 𝐶 prime, corresponds with the line 𝐵𝐶 of length three on the smaller triangle. So, we can calculate 𝑦 by multiplying the length three by the scale factor of five over two. Three times five is 15, and 15 over two simplifies to 7.5. And so, our answer to the second part of this question is 𝑦 equals 7.5.
There is an alternative method we could’ve used to work out the value of 𝑦. As we know that our triangles are similar, our lengths will all be in the same proportion. Looking at the lengths 𝐴 prime 𝐵 prime, which is five, and 𝐴𝐵, which is two, we can say that five over two is equal to 𝑦 over three. As these triangles are similar, we know that it’s the same proportion between the lengths five and two as it would be between 𝑦 and three. We can then take the cross product, and so two times 𝑦 is two 𝑦 equals five times three, which is 15. And if two 𝑦 is 15, then 𝑦 is half of that. So, 𝑦 is 7.5, confirming the answer that we found by working out the scale factor.
Let’s take a look at the final part of this question to find the value of 𝑧. We know that we go from the smaller triangle to the larger triangle by multiplying the lengths by five over two. But what happens in the reverse direction? In this case, we’d have to perform the inverse operation. We could informally say that we need to divide the lengths by five over two, but scale factors should be given as a multiplier. We can recall that when we’re dividing by a fraction, this is equivalent to multiplying by the fraction flipped. So, our scale factor from the large triangle to the small triangle would be two-fifths.
In order to work out the length of 𝑧 on the line 𝐴𝐶, we take the corresponding length 𝐴 prime 𝐶 prime, which is four, and multiply it by the scale factor of two-fifths. Four times two is eight, and eight-fifths is equivalent to 1.6. So, our answer for the final part of this question is that 𝑧 is 1.6.
We could’ve worked out this final question by using the original scale factor. We would’ve set up an equation that said 𝑧 times five over two equals four. We would’ve then rearranged this to find the value of 𝑧. This method does track very closely to finding the reverse scale factor however. Both methods would confirm that 𝑧 is 1.6.
Before we summarize what we’ve learned in this video, let’s take a closer look at reverse scale factors. Let’s say that we’re given a pair of similar triangles and we’re told that the scale factor from the smaller triangle to the larger triangle is three. Is there a quick way to find the scale factor in the reverse direction? Well, the scale factor in the reverse direction will always be the reciprocal, that is, one over the number. The reciprocal of three is one-third.
If the scale factor in one direction is six, then the scale factor in the reverse direction will be one over six. So, this will be one-sixth. How about a fractional scale factor like twelve-sevenths? Well, the reciprocal of that is the fraction flipped. So, the scale factor in the reverse direction would be seven-twelfths. Using these facts on reverse scale factors will help us to find any lengths on similar triangles.
Now, we can review what we’ve learnt in this video. We saw that triangles are similar if all the corresponding pairs of angles are congruent and corresponding sides are in proportion. We can prove that triangles are similar by using either the AA rule, the SSS rule, or the SAS rule. It’s very important to remember that we must show that the angles are congruent and sides are in proportion. The SSS and SAS rules may be familiar from when we’re showing that triangles are congruent. However, when we’re showing that triangles are congruent, we’re showing that the sides are congruent. When we’re showing these rules in similarity, we’re showing that the sides are in proportion.
The scale factor is the ratio of corresponding sides in similar triangles. It is always given as a multiplier. We can find the scale factor in a particular direction by calculating the new length of a side divided by the original length of the side. And finally, to find a reverse scale factor, we find the reciprocal of the scale factor.
