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Hello, my name is Dr.

Rollinson and I'm excited to be guiding you through today's lesson.

Let's get started.

Welcome to today's lesson from the unit of trigonometry.

This lesson is called "Checking and Securing Understanding of Similar Triangles" and by the end of today's lesson we'll recognise that similar shapes have sides in proportion to each other, but the angle sizes are preserved.

Here are some previous keywords that we're going to use again during today's lesson, so if you want to remind yourselves what any of these words mean before we start, you may want to pause the video while you do so and press play when you're ready to continue.

This lesson contains two learn cycles.

In the first learning cycle, we're going to focus on how to identify whether or not a pair of triangles are similar, and then in the second learn cycle, we're going to look at cases where we have a pair of triangles that we know are similar and we're going to use that fact to find some missing lengths.

Let's start off with "Identifying Similar Triangles".

Here we have a triangle with the lengths and angles displayed.

The triangles about to be enlarged by a scale factor of two to create a similar shape.

Andeep, Izzy and Sofia are considering which properties the triangle they think will change after it's been enlarged.

Let's hear from them now.

Andeep says, "I think the angles will change." Izzy says, "I think the lens will change", and Sofia says, "I think the proportions between the lengths will change."` What do you think? Out of those three statements, who do you agree with? Pause the video while you think about it and impress play when you're ready to continue.

Out of these three statements, what Izzy said is correct, but the other two statements are not correct.

Let's take a look at this in a little bit more detail.

When a shape is enlarged, the angles do not change.

For the example we can see on the screen here, this factor is probably easiest to see with the right angle.

With the triangle on the left hand side, the bottom right vertex is a right angle and with the triangle on the right hand side, even though that triangle is bigger in size in terms of its lengths, the vertex in the bottom right hand corner, it's still a right angle and the other two angles are the same as well.

Therefore, similar shapes have the same angles.

On the other hand, when a shape is enlarged, the lengths may change.

This does depend on the scale factor though.

For example, here we have lengths three metres, four metres, and five metres on a left hand triangle.

If we enlarge that by a scale factor of two, the edges on the right hand triangle will all be twice the length of the edges on the left hand triangle.

We have six metres, eight metres and 10 metres, so the lengths have different there.

Therefore similar shapes may have different lengths.

When a shape is enlarged though the proportions between each pair of lengths do not change.

The way we can see this is by taking each possible pairing of lengths, dividing one length by the other to see what the multiplier is between them and see if they are the same in both triangles.

For example, one possible pairing of lengths we could do on the left hand triangle is three metres and five metres.

We could divide three by five to get naught 0.

6 and that means five times naught 0.

6 is three.

Let's do the same with the correspondent edges on the right hand triangle.

We do six divided by 10, we get naught 0.

6 as well, same results.

What about the other pairings? We could do four divided by five to get naught 0.

8 and we could do eight divided by 10 to get naught 0.

8 as well.

Same result, and then there's only one pairing left.

We could do three divided by four to get naught 0.

75, and if we do the same on the other triangle, six divided by eight, we also get naught 0.

75.

So we have the same multipliers between each pair of length in both the triangles.

Therefore similar triangles have lengths in the same proportions.

Now you may already know some criteria about how to prove two triangles are congruent.

That criteria can also be used as well to show that any unknown lengths or unknown angles in a triangle are fixed by measurements that are known.

For example, with this per triangles here, we know all three lengths and all three lengths are the same, but even though we don't know the angles, we know that the angles in the top triangle must be the same as angles in the bottom triangle because they are fixed by the lengths.

In these examples, we know two angles and a length of the side between them and we can see that they are the same.

Even though we don't know the two remaining lengths of each triangle, they are fixed by the information that we have.

And the third remaining angle, we could work that out if we want to by subtracting from the 180 degrees and we'll see, that they are the same as well.

We might know two angles on a side in this configuration.

Once again, we don't know what the two remaining lengths are, but they are fixed by the information that we have here and so is the remaining angle.

On this case, we have a right angle, we know the length of the hypotenuse and the length of one of the shorter sides.

That information is enough to determine that the two angles we don't know and the length we don't know are fixed.

There are not different options for what they could be.

There is only one option for what they could be, even if we don't know what they are.

Now in these examples here, we can see that each triangle is congruent to the one above or below it, but this criteria can also be adapted to determine similarity between triangles.

The only difference here is rather than the lengths all being the same, the lengths will be different, but they'll be in the same proportions.

That means for any cases where we have two or more lengths in a triangle, if we divide one of those lengths by the other, we should get the same result as if we do it with the correspondent lengths in the similar triangle.

So for example here, the multiplier between five and three is the same as the multiplier between 10 and six.

The multiplier between five and four is the same as the multiplier between 10 and eight, and the multiplier between four and three is the same as the multiplier between eight and six.

So we can take all the rules we've learned about congruency and adapt them slightly to use for similarity, with a difference being for the lengths may differ, but they are in the same proportions.

With similar triangles, we can also prove similarity in another way, but we can't necessarily with congruence.

And that is similarity with triangles can also be determined by all three of the angles.

Here we have two similar triangles and we can see that the angles are the same and we can do that by overlaying one triangle on top of the other at each of its vertices.

Like so.

So when two triangles are similar, they have the same angles and when two triangles are the same angles, they are similar.

So let's take a look at a few examples together now.

Here we have a pair of triangles and Aisha and Lucas are trying to work out if there is enough information here to determine that these triangles are similar.

Aisha says, "How could we determine similarity without knowing whether the angles are the same?" Pause the video while you think about whether or not it is possible to determine whether these triangles are similar based on the information we have.

And then press play when you're ready to continue.

Let's hear from Lucas then.

Lucas says, "We could find the proportions between each pair of lengths on one triangle." So let's do that.

We could do five divided by eight to get naught 0.

625.

Four divided by eight to get naught 0.

5, and four divided by five to get naught 0.

And then we could check if those proportions can be found on the other triangle.

Now we could do that by dividing each per lengths in the same sort of way or we could take the multipliers that we've just worked out and see if we can apply those to between pairs of lengths on the other triangle.

So for example, naught 0.

625, well, 12 times naught 0.

625 is 7.

How could we use a naught 0.

5? 12 times naught 0.

5 is six.

And then can we see where that naught 0.

8 fits? 7.

5 times naught 0.

8, is six.

So the lengths of the triangles are in the same proportions, so the angles must be the same and therefore they are similar.

Here's another pair of triangles.

Is there enough information to determine that these triangles are similar? Jacob says, "How could we determine similarity without knowing any of the lengths?" Pause the video while you think about this and press play when you're ready to continue.

Sam says, "The two triangles have the same angles, therefore the lengths must be in the same proportion, so they are similar." If the angles are the same in two triangles, you know they're similar.

So how about in this situation then? We have two of the angles in each triangle.

Jacob says, "We only know two the angles, so can we still determine similarity?" Pause the video while you think about this and press play when you're ready to continue.

Sam says, "We can see two of the angles are the same in each triangle.

That means the third angle must also be the same.

We could double check that by finding the missing angle in each." But either way, we know the triangles are similar, because we know the angles are the same.

So how about this situation? We have two triangles here where we know they are right angle triangles and we have two of the lengths in each.

Is there enough information to determine similarity here? Pause the video while you think about it and press play when you ready to hear from Jun, Alex and Laura.

Well let's hear what they have to say.

Jun says, "We have a right angle and we know the lengths of the hypotenuse and another side." So Jun's thinking here about the congruence rule, where you have a right angle, the hypotenuse and a side length.

Alex says, "If the lengths were the same then we could prove that they're congruent." Laura says, "That would mean that the triangles have the same angles and there's only one possible length for the third edge." That's what it means when two triangles are congruent, they have the same angles and the same lengths.

However, in this case we can see that they're not congruent, because the lengths are different.

Jun says, "The two known lengths of the large triangle are both double the correspondent length on the small triangle." So we know that those two lengths are the same proportion.

But what about the third length? Alex says, "We could use the right angle hypotenuse side criteria to establish that there's only one possible length for the remaining edge in the large triangle." "Therefore," Laura says, "this edge must also be twice the length of its correspondent edge." We don't necessarily know how long that horizontal edge in a small triangle is, but if we call it "X", then in the large triangle that must be 2x.

It is twice the length, so all three of 'em now say "The triangles are similar." Let's check what we've learned.

Here, we have a pair of triangles and we have three statements.

Which of these three statements is true? A, the triangles are similar.

B, the triangles are not similar.

And C, there is not enough information to determine similarity.

Pause the video why you select which one you think is true and press play when you're ready for an answer.

The answer is A, the triangles are similar.

We have two angles and a length in each triangle.

Now the easiest way to determine in this case is to find the third remaining angle and we'll see that the angles in both the triangles are the same as each other, therefore they are similar.

And here's another pair of triangles, but we have the same three statements.

Which of these statements is true for this per triangles? Pause video while you make a choice and press play when you're ready for an answer.

The answer is A, the triangles are similar.

We have a right angle triangle with the length of the hypotenuse and a short side, so we can apply the criteria from congruence, but the lengths aren't the same, so we just need to double check are they in the same proportions? 12 divided by six is two 18 divided by nine is two, or we could do six divided by 12 is naught 0.

5 and nine divided by 18 is naught 0.

Either way, we can see they're in the same proportions.

Which statement is true this time? Pause the video while you make a choice and press play when you're ready for an answer.

The answer is C.

There is not enough information to determine similarity.

We have the lens of two of the sides and one of the angles, but the not in the configuration where we can be absolutely sure that the third remaining length or the two remaining angles are fixed.

If the angle was between those two lengths, that would be fine, but they're not.

Okay, it's over to you now, for task A, this task contains one question and here it is, we have a triangle, A, B, C, with its three lengths displayed and two of its angles and we have seven other triangles.

For each of those triangles, you need to compare it to triangle ABC, write a tick in any triangles where you know it's similar to ABC, write a cross in any triangles where you know it's not similar to ABC, and write a question mark in any triangles where does not enough information to determine similarity.

Now the triangles are not drawn accurately, so using a protractor and a ruler will not help you in this case and also all lengths are given in the same units.

Pause the video while you work through this and then press play when you're ready for an answer.

So let's see how we got on with that.

In part A, this triangle is similar to triangle ABC.

We have two lengths and the angle that is between them, so that's enough to know that the remaining measurements that we don't know are fixed.

We can also see that if we do 44 divided by 55, we'd get the same result as if we divided 40 by 50.

So lengths are in the same proportions.

Part B, this triangle is also similar to ABC.

We don't know any angles, but, if we take each per lens and do a division, 28 divided by 35 and 28 divided by 21 and so on, we'll get the same results as if we did those divisions on triangle ABC.

In part C, this triangle is not similar to ABC.

Yes, we have a length and angle and a length, in that configuration, but those lengths are not in the same proportions as with triangle ABC.

If we do 32 divided by 45, we do not get the same result as 40 divided by 50.

In part D, this triangle is similar to ABC.

We don't know any lengths, but we know two of the angles in each triangle and we know those angles are the same.

The third angle must also be the same as well.

If you wanted to double check, you could subtract from 180 but you don't necessarily need to.

It's enough to know that they are similar.

In part E, this triangle is not similar to ABC.

We have a right angle and we have two of the lengths and if you do 48 divided by 60, you do get the same result, as if you divide 40 by 50.

But the 48 and the 60 in part E, correspond with the 30 and the 40 in triangle ABC, 'cause those are the edges which are adjacent to the right angle.

And you do 40 divide by 60, you don't get the same as 30 divide by 40.

In part F, this triangle is similar to ABC, even though you've got the right angle and the same two lengths, 48 and 60, those lengths are in the correct configuration compared to ABC.

If you do 48 divided by 60, you get the same result as 40 divided by 50.

And then in part G, this triangle does not contain enough information to determine similarity.

You have an angle and two lengths, but they're not in a to determine that the remaining length and the remaining two angles are fixed.

Great work so far.

Now let's move on to the second learning cycle, which is "Finding missing length with similar triangles." Here we have a pair of triangles and these two triangles are similar.

If we wanted to prove that, well there's a couple of ways we could do it, but the more straightforward way would probably be to find a missing angle in each triangle and show that all the angles are the same between these two triangles.

Now we know one length in each triangle.

If we knew another length in one of the triangles, then it's corresponding length in the other triangle could be calculated through proportional reason.

For example, we know the length of the hypotenuse in each of these two triangles.

In the left hand triangle we can see that the vertical length, which is opposite the 30 degrees is eight metres.

The length which is opposite of 30 degrees on the right hand side is labelled Y.

We could work out Y in a couple of different ways.

One way is to look at a scale factor to get from the left hand triangle to the right hand triangle.

We could do 20 divided by 16 to get 1.

25 as our scale factor, then apply that to the eight.

Eight times 1.

25 is 10.

A second method we could do, could be to look at the proportions within each triangle.

We could look at how do you get from 16 to eight.

We could do eight divided by 16 to get naught 0.

So we apply that same multiplier to the right hand triangle.

We do 20 times naught 0.

5 to get 10 again for our value of Y.

Scale factors and multipliers within triangles can be easy to find where there is an edge with a length of one unit.

For example, here we have a pair of similar triangles.

We know the similar because the angles are the same.

In the left hand triangle we have a hypotenuse with length one unit and in the other two lengths are decimals.

Then the other triangle, we have a hypotenuse of 20 metres.

If we want to work out the values of X and Y, which are the other two lengths, we could do that using again, two different methods.

One method could be to get the scale factor.

Now the reason why it is easier with having one of the lengths as being one unit, is that we don't really need to divide one length by another to figure out what the scale factor is.

We know that one times 20 is 20.

And then we could use that scale factor to work out X and Y.

Naught 0.

875 times 20 is 17.

4, naught 0.

5 times 20 is 10.

A second method could be to look at the proportions again within each triangle.

And once again this is made easier by the fact that one of the length is one unit, because we don't need to do a division between naught 0.

87 and one to figure out what the multiplier is.

We know that one times naught 0.

87, is naught 0.

87, so we could do the same on the other triangle.

20 times naught 0.

87 gives you a value of X, of being 17.

4, and then the other multiplier to get from one to naught 0.

5, that's just times naught 0.

So let's do 20 times naught 0.

5, to get a value of Y of 10.

And here we have another pair of similar triangles.

Once again, we know that the triangles are similar, because the angles are all the same.

As one of the triangles has an edge with length one unit.

It may be easier to spot the proportion relationships within the triangles, rather than calculating the scale factors between them.

And the reason why is that the known lengths which correspond with each other between these two triangles do not have a length of one unit.

The length of one unit in the left hand triangle corresponds with our unknown X in the other triangle.

So let's look at the proportions within instead.

On the left hand triangle to get from one to naught 0.

5, we just times by naught 0.

So with the right hand triangle, X times naught 0.

5 must be 12.

So if I want to work out the value X, I could use the inverse operation, could do 12 divided by naught 0.

5 to get 24.

So let's check what we've learned.

Here we have a pair of triangles which we can see are similar.

You need to find the value of Y, pause the video while you have a go and press play when you're ready for an answer.

Okay, let's take a look at the answer using two methods.

Using a scale factor method would get a scale factor of 1.

2, which gives a value of Y of nine.

And if we use the proportions within each triangle, well to get from the hypotenuse to the length or the edge that is opposite the 49 degrees, it is times naught 0.

75.

And we do that on the right hand triangle, we get again a value of nine for Y.

And how about for this per triangles, find the value of X this time.

Pause the video while you have a go and press play when you're ready for an answer.

Once again, let's look at the answer in two different ways.

Method one could be to find the scale factor between them, which will be 12 and our value of X will be 8.

Method two, use the proportions within each triangle to get from the hypotenuse, which is one to the length, which is between the two angles, which is naught 0.

We times by naught 0.

Do the same other triangle once again we get 8.

4 for our value of X.

And then one more time, what is the value of X this time? Pause the video while you have a go and press play when you're ready for an answer.

Well we can see that the proportions within the triangle on the left hand side are already given to us, so we can use that on the right hand triangle to find a value of X.

X times naught 0.

7 gives you 12.

So we can do 12 divided by naught 0.

7 to get 17.

So the key fact we've used multiple times so far, is that when a triangle's enlarged, the multipliers between pairs of lengths remain the same.

So if we take this triangle we can see here, where it has a length of one for the hypotenuse and the side which is opposite the 30 degree angle is naught 0.

5, we can see the multiplier to get from one number to the other is times naught 0.

If we enlarge triangle, so the hypotenuse is two, the length which is opposite a third degree angle, would be one and it's the same multiplier between them.

And we enlarge it again, we can see that that multiplier between the hypos and the length, which is opposite the third degree angle, remains the same each time.

It's naught 0.

And this can be seen when we have triangles that are nested inside each other like the left hand diagram.

And when the triangles are drawn separately, like in the right hand diagram.

The right hand diagram is the same as the left hand diagram, but I've taken a small triangle out of it and placed it separately.

So with that in mind, a pair of similar triangles can be made from a single triangle by taking a measurement partway along some of its edges.

For example, here we have a right angle triangle where it has a length of 3.

9 metres and we want to know what is the length of the hypotenuse.

Now we can't do that with the information we have here alone, but what we could do is take a measurement partway along this by drawing a line that is parallel to one of the edges of the triangle.

In this case, we're drawing a line that is parallel to the vertical edge of the triangle and if we take measurements from the left vertex to where this line intersects the other two edges, we can use that to help us.

In this case, the line has been drawn one metre along the hypotenuse and then looking along the horizontal edge, we can see it's naught 0.

75 metres across.

So now we've got that information, we can create a pair of similar triangles.

And what's quite helpful here, is that for one triangles there is a length that is one and that makes it easier to find the proportions within in each triangle because one times naught 0.

75 gives you naught 0.

75.

So x times naught 0.

75 gives you a 3.

Therefore we do 3.

9 divided by naught 0.

75 to get the value of X, which is 5.

27.

Here we have Sam in a field.

In the field there is a tall pole that is held upright by some ropes and Sam wants to work out the height of the pole.

Now Sam says I can't measure all the way up the pole or all the way along the rope.

She just can't reach that far.

But what she can do is measure the distance along the floor.

So the distance from the foot of the pole to where one of the ropes meets the floor is eight metres.

Now bearing in mind that Sam has some equipment for measuring, but she just can't reach as high as the top of the pole, how could similar triangles help Sam with this problem? Pause the video while you think about this and press play when you're ready to continue.

Well, Sam says, "I could measure one metre along the ground from where the rope meets to floor, and then at that point I could measure the vertical distance between the rope and the floor.

Then I have a pair of nested similar triangles, which I could use to find the height of the pole." And the reason why Sam used one metre as her first measurement, is that it makes this multiplier easy to see.

One times naught 0.

75 is naught 0.

75.

Therefore if we multiply eight by naught 0.

75, we'd get the height of the flag pole.

The pole is six metres tall.

Here we have Jacob inside his attic.

Jacob wants to estimate the length of his roof from inside the attic.

Now a problem that Jacob has here is he can't measure the length of the roof from inside the attic because not all of it is available.

He can't even measure all the way across the floor to the edge of the roof because some of it is boxed off on the left.

But what he has done is he's measured the vertical distance from the top of the roof down to the floor of his attic.

So how could similar triangles help Jacob with this problem? Pause the video while you think about this and press play when you are ready to work through it together.

Well, sometimes when you're working with practical situations it can be helpful to draw a diagram.

Here we have the photo on the left and with a diagram on the right and the length of we're trying to work out has been labelled X.

So what could Jacob do? Well, a length of one metre could be measured along the roof from the top, like so.

If we label it onto our diagram, it would be here.

And then the vertical height from here to the top could also be measured as well, which is naught 0.

3, two metres.

And we've labelled that on our diagram as well.

And then from this point, similar triangles could be used to find the length of the roof.

We have a small triangle nested inside a larger triangle.

And those two triangles are similar.

And once again, the reason why a length of one metre was used was because it makes the proportions easier to see.

One times naught 0.

32 is naught 0.

32.

So X times naught 0.

32 would be 2.

And that means 2.

3 divided by naught 0.

32 will give us the value of X, which is 7.

19.

So let's check what we've learned.

Here we have a diagram with some measurements labelled on, what is the value of x? Pause video while you work it out and make a choice and then press play when you're ready for an answer.

The answer is D, five is our value of X.

What is the value of X this time? Pause video while you work it out and press play When you're ready for an answer.

The answer is C, four.

The entire length of that hypotenuse is five metres.

We'd subtract one metre to get the length of what X is.

And then here's a slightly different diagram.

What is the value of X this time? Pause the video while you work it out and press play When you're ready for an answer.

The answer is C, four metres.

Okay, it's over to you now, for task B, this task contains two questions and here is question one.

We have eight right angle triangles, which all have an angle of 24 degrees, which means that they all are similar.

For one of the triangles you can see all three of its lengths and one lengths is one unit.

And then in all of the other triangles you need to work out the unknown lengths that are labelled on those triangles.

Pause video while you have a go at this and press play when you're ready for question two.

And here is question two.

You have two diagrams here, with unknowns X, Y, and Z labelled somewhere on these diagrams. You need to find the values of X, Y, and Z.

Pause the video while you do this and press play when you're ready for some answers.

Okay, let's see how we got on.

In question one.

A is 1.

35.

B is naught 0.

C is 1.

8, D is naught 0.

E is one, F is 1.

62, G is two, H is 4.

5, I is three, and J is 2.

How did we get on with that? And in question two, the value of X is 6.

The value of Y is naught 0.

98 and the value of Zed is two.

Fantastic work today.

Let's now summarise what we've learned in this lesson.

An enlargement is a transformation that causes a change in size and that could be a change in size that makes it bigger or it could make it smaller.

But when an object is enlarged, the image is similar.

That means the lengths may change, but the proportions will remain fixed in each shape, and the angles also do not change.

And just as with congruent triangles, not all lengths and angles need to be known to determine whether or not a pair of triangles are similar.

If two triangles have the same angles, they are also similar as well.

And the proportional relationship between similar triangles and within triangles can be used to find some missing lengths like we've done in the second part of today's lesson.

Thank you very much.

Have a great day.

I've finished the video