### 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.