Lesson video

Hello, my name is Dr.

Rowlandson and I'll be guiding you through today's lesson.

Let's get started.

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

This lesson is called describing an enlargement.

And by end of today's lesson, we'll understand the minimum information that is required to describe an enlargement.

The lesson will be introduced in two keywords.

The first keyword is scale factor, which is the multiplier between similar shapes that describes how large one shape is compared to another.

And the other is centre of enlargement, which is a point from which a shape is enlarged.

The lesson contains two learn cycles with each learn cycle, exploring on those keywords in more depth.

And to begin with, we'll be starting with using and finding scale factors.

Here, we have an object and an image where the object has been enlarged to create the image.

How could we describe the change in size? Let's look at two descriptions and compare them.

Laura says, "It has got four centimetres longer "in one direction and 10 centimetres longer "in another direction." Whereas Andeep says, "The lengths of the image "are three times the lengths of the object." So Laura's description has used addition to describe the change in size and Andeep's description has used multiplication to describe the change in the size.

Let's compare them.

Laura's description, one problem with it is that each length requires a different addition.

Two centimetres plus four centimetres makes six centimetres and five centimetres plus 10 centimetres makes the 15 centimetres.

If this shape had more sides, which are all different lengths or require more and more additions to describe the change in size with a different addition for each different length, whereas Andeep's description using multiplication doesn't have that problem because all the sides have been multiplied by the same thing.

Two times three is six and five times three is 15.

A scale factor is a multiplier between similar shapes that describes how large one shape is compared to another.

So in this case, the scale factor is talking about the multiplication from two and five to get six and 15.

So for example, two centimetres times three makes six centimetres, and five centimetres times three makes 15 centimetres.

An enlargement can be described by stating the scale factor from the object to the image.

And in this case, the scale factor is three.

The scale factor can be calculated by dividing a length from the image by its corresponding length from the object.

The scale factor is the same for all corresponding lengths because the shapes are similar.

For example, to get the scale factor here, we can divide six by two to get three.

And to get the scale factor here from five to 15, we can divide 15 by five to get three.

So the scale factor here is three to get from the object to the image.

In some cases, you will not necessarily think explicitly about these divisions because you might spot what the scale factor is based on your times table knowledge.

We know that two times three is six and five times three is 15.

Therefore, the scale factor must be three.

However, not all scale factors are whole numbers and what all necessarily be as easy to work out through times tables.

So in those cases, we will need to use divisions to figure out what the scale factors are.

For example, the scale factor can be a fraction.

In this example, the object is larger than the image.

So if I wanna get this scale factor from the object to the image, what I multiply six by to get two? We can get it by doing two divided by six, image divided by object, which gives one third.

And we can do five divided by 15, image divided by object, to get also one third.

So in this case, the scale factor is one third.

Let's check what we've understood here.

Calculate the scale factor for the enlargement from the object to the image.

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

The answer is four.

We calculate it by doing 20 divided by five, the image divided by the object to get four.

Let's try again.

Calculate the scale factor for this enlargement from the object to the image.

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

In this case, the image is smaller than the object, so we need to be careful about which way round we do the division.

We're gonna do four divided by eight, image divided by object.

Four divided by eight is one half.

Let's now look at using scale factors.

The scale factor for an enlargement can be used to calculate missing lengths in the image.

For example, here, we have an object which is a parallelogram with lengths two centimetres and six centimetres, and we have an image which is a similar parallelogram with length eight centimetres and a missing length which is labelled x.

We can work out the missing length on the image by first calculating the scale factor between these two shapes, and then using that scale factor to calculate the missing length.

We can calculate the scale factor by using these two corresponding lengths and thinking what do times two by to get eight or we can calculate it with the division eight divided by two, which is four.

If that multiplier is four, then this multiplier must also be four.

Six times four makes 24.

Therefore, the missing length is 24 centimetres.

Let's look at another example.

Here, we have two similar triangles where this time, the image is smaller than the object and we need to work out the missing length x at the base of the image.

This question also has an additional challenge in that the image contains more information than we necessarily need.

So we need to select the right information to calculate the scale factor.

To calculate the scale factor, we need to use two corresponding lengths.

The 12 and the four correspond with each other.

So we need to do four divided by 12 to get a scale factor of one third.

And then to calculate the missing length, we'll do 15 times one third, which makes five centimetres.

So the missing length is five centimetres.

Let's check what we've learned.

The object has been enlarged to create the image.

What is the value of x? Is it A.

Four, B.

16, C.

20, or D.

25? Pause the video, have a go, and press play when you're ready for the answer.

The answer is B.

16.

We can get 16 by first calculating the scale factor from doing 20 centimetres divided by five centimetres to get a scale factor of four and multiply that scale factor of four by the length of four centimetres to get 16.

If you put A as your answer, it may be that you calculated the scale factor correctly, but then put that as your answer rather than using the scale factor to work out a missing length.

If you put C.

20, it might be that you correctly calculated the scale factor of four, but then multiply that four by the length of five centimetres rather than the length of four centimetres.

And if you put D, it might be that you incorrectly calculated the scale factor because you divided the 20 centimetres by four centimetres rather than the 20 centimetres by five centimetres.

Remember, you must divide corresponding lengths to get a scale factor and the 20 corresponds with a five.

Here's another one, the object has been enlarged to create the image.

What is the value of x? Is it A.

five, B.

10, C.

20, or D.

90? Pause, have a go, and press play when you're ready for an answer.

The answer is A.

five.

We can get that by first calculating the scale factor from doing 10 millimetres divided by 60 millimetres to get a scale factor of one-sixth, and then multiply the 30 millimetres by one-sixth to get five millimetres.

So the value of x must be five.

If you put 10, it might be that you correctly calculated the scale factor of one-sixth, but then you multiplied the length of 60 millimetres by one-sixth rather than the length of 30 millimetres.

If you put C.

20, it might be that you incorrectly calculated a scale factor because you divided the 10 millimetres from the image by the 30 millimetres from the object.

But remember, we must divide corresponding lengths to calculate the scale factor and the 10 millimetres of the image corresponds to the 60 millimetres in the object.

And if you put D.

90, it might be that you've calculated your scale factor the wrong way around by doing the object divided by the image and multiplied it by one of the lengths.

Okay, over to you.

For Task A, this task contains two questions.

In question one, we can see parallelogram P, which is in the top left corner there, has been enlarged to create parallelograms A, B, C, and D.

And in each one, we need to calculate the missing lengths on those parallelograms. Pause the video, have a go, and press play when you're ready for question two.

And here's question two.

Take the diagram that shows the correct enlargement of the small triangle.

So we've got a small triangle at the top of each of those three diagrams and it's been enlarged to create the larger triangle underneath each time, but only one of them is correct.

Which one is correct? Once you've done that, here's something else to think about.

Sam draws another enlargement of the small triangle.

One of the lengths is 12 centimetres.

What could the other length be? Post a video video, have a go, and press play when you're ready for the answers.

Okay, let's see how we got on.

So in question one, we need to calculate the missing length in each parallelogram.

In A, the missing length is 16.

The scale factor from nine to 18 centimetres is two.

So we need to multiply the eight by two to get 16.

In B, the missing length is four.

We get the scale factor from doing the three centimetres in the image divided by the six centimetres in the object as a scale factor of one half, and then multiply that half by the eight centimetres in the object to get four centimetres.

In C, the missing length is three centimetres.

We get a scale factor again by doing the two centimetres in the image divided by the six centimetres in the object, so that is a scale factor of one third, and then times the nine centimetres from the object by a third to get three centimetres in the image.

And in D, the missing length is 15 centimetres.

In this case, we need to calculate the scale factor from doing 20 centimetres in the image divided by eight centimetres in the object which is 2.

5 or two and a half, and then we need to multiply the six centimetres from the object by two and a half to get 15 centimetres in the image.

In question two, we had to tick the diagram that shows the correct enlargement of the small triangle.

So the small triangle is the object and the large triangle is the image.

Let's look at the object in more detail.

The object has a horizontal length of two centimetres and a vertical length of three centimetres.

And in the image, we can see that in each triangle, the horizontal length is six centimetres, which would imply that the scale factor must be three to get from two to six to get three.

But the vertical length is different in each one.

In the first one, the vertical length is seven.

In the second one, it's eight, and the third one, it's nine.

So which is correct from those? The object has a vertical length of three, the scale factor is three, three times three makes nine.

So the correct answer is the third one with a vertical length of nine centimetres.

And then Sam draws another enlargement of the small triangle.

One of the lengths is 12 centimetres.

What could the other length be? Here are two possible answers we could have for this question, eight centimetres and 18 centimetres.

The reason why there are multiple answers to this problem is because it doesn't tell us in the question which length is 12 centimetres in the image.

It could be the horizontal length or the vertical length, which is 12.

If it is the vertical length which is 12 centimetres, then we'll be going from three centimetres in the object to 12 centimetres in the image as a scale factor of four.

And then we do two times four to get the eight centimetres.

But if it's the horizontal length in the image that is 12 centimetres, we'll be going from two centimetres in the object to 12 centimetres in the image.

That's a scale factor of six.

12 divided by two is six.

And then we need to multiply that by three to get the 18.

So it could be either eight or 18 centimetres.

Alternatively, there is a third length in that triangle.

If we measured the diagonal length for that triangle to be 12 centimetres, and then calculated the other two lengths, they would be approximately, but not exactly 6.

7 centimetres and 10 centimetres.

So any of those answers are correct.

Well done, you're doing brilliantly.

Let's now look at how this centre of enlargement affects the image.

Here, we have a grid with a triangle on for the object.

And Aisha is going to enlarge the object by a scale factor of two, but she is not sure where to place the image on the grid.

The centre of enlargement is a point from which the object is enlarged.

So where that centre of enlargement is affects where the image goes.

So that means the scale factor not only affects the lengths of the image, but also affects how far each vertex is away from the centre of enlargement.

In this case, the scale factor between the object and the image is two.

And if we look at the object and the vertex at the very top of that triangle, we can see that that vertex is five units away from the centre of enlargement, whereas the same vertex in the image is 10 units away from the centre of enlargement.

So the distance of five from the object's vertex, the centre, has also been multiplied by two to get the distance of 10 from the image to the centre.

The position of the centre of enlargement defines where the image will be placed in relation to the object.

As that centre of enlargement moves around the grid, so does the image as well.

The centre of enlargement can be found by drawing lines through corresponding vertices of the object and the image.

For example, here, we have two similar triangles.

It doesn't really matter which ones the object and the image.

If we want to find the centre of enlargement, we can do it by drawing a straight line through a vertex of one and its correspondent vertex in the other, and then doing that for each pair of correspondent vertices.

And we can see that those three lines all intersect at the same point.

The point where the lines intersect is the centre of enlargement, that's where the shape has been enlarged from.

The centre of enlargement can also be described as a pair of coordinates.

For example, find the coordinates of the centre of enlargement in this case.

We can do it, again, by drawing straight lines through pairs of correspondent vertices.

And the point where those three lines meet is the centre of enlargement, which in this case, is the coordinate (6,7).

The centre of enlargement can be anywhere on or off the object.

In all the examples we've seen so far, the centre of enlargement has been off the object, outside the object, but it can also be somewhere on the object too.

That point will always remain invariant in the image.

For example, the centre of enlargement here is outside of the object and outside of the image.

In this example, the centre of enlargement is in the middle of the edge for the object and in the middle of the edge in the image as well.

That point is invariant.

In this example, the centre of enlargement is in the top left vertex of the object, which also means it's in the top left vertex of the image and that vertex is invariant.

And in this example, the centre of enlargement is in the very centre of the object, which means it's in the very centre of the image and that point is invariant.

Let's check what we've learned.

Which diagram has correctly drawn a line to locate the centre of enlargement? Is it A, B, or C? Pause the video, have a go, and press play when you're ready for the answer.

The answer is C.

In A, we can see the line has not gone through any vertices.

In B, we can see the line has gone through the bottom left vertex of the large triangle, but the top vertex of the smaller triangle, no, that line needs to go through correspondent vertices of each triangle, which is what it does in C.

Another question, if the centre of enlargement in this case moved to the right, in which direction would the image move in? Is it A.

Left, B.

Right, C.

Up, or D.

Down? Pause the video, have a go, press play when you're ready for the answer.

The answer is A.

Left.

As I move the centre of enlargement to the right, we can see that the image moves to the left.

And a third question, at which point is the centre of enlargement? So we can see, we have two triangles, the small one and the large one.

And it's been enlarged from either point A, point B, point C, or point D.

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

The answer is C.

It's been enlarged from point C, which is in the middle of the edge on the both the small triangle and the large triangle.

A full description of an enlargement requires both the scale factor and the centre of enlargement.

We need to state the scale factor so we know how much bigger or smaller the image has got and we need to state the centre of enlargement so we know the position of the image in relation to the object.

For example, fully describe a transformation from A to A prime in this diagram.

First, let's get the scale factor and we can do that by comparing corresponding length.

The image has a width of three and the object has a width of one.

So the scale factor is three divided by one, which is three.

Now, let's get the centre of enlargement, which we can do by drawing a line through corresponding vertices on the image and the object.

We can see that all follow those lines intersect at the same point which is at the coordinate (11,4).

Therefore, the centre of enlargement is at (11,4).

We have now all the pieces we need to fully describe this transformation.

Object A has been enlarged by a scale factor of three from the point (11,4).

Let's check what we learned from there.

Which information is missing from this description below? The description says object A has been enlarged by a scale factor of two.

What information is missing? Is it A.

The centre of enlargement, B.

The line of symmetry, C.

The scale factor, or D.

The vector.

Pause, have a go, press play when you're ready for the answer.

The answer is A.

The centre of enlargement is missing from that description.

Which information is missing from this description below? Object A has been enlarged from the 0.

23.

Which is missing? Is it A.

The centre enlargement, B.

The line of symmetry, C.

The scale factor, or D.

The vector.

Pause, have a go, press play when you're ready for the answer.

The answer is C.

The scale factor is missing.

Over to you now for Task B.

This task contains two questions.

The first question, we need to tick the enlargements that have the same centre of enlargements.

So we can see there are three pairs of enlargements, A to A prime, B to B prime, and C to C prime.

Two of those have the same centre of enlargement and one of them is different.

Take the two that have the same centre of enlargement.

Pause the video, have a go, and press play when you're ready for question two.

Question two, describe each transformation fully.

In each case from A to A prime and B to B prime, we need to describe it with all the details that are required to describe a transformation of this kind.

Pause the video, have a go, and press play when you're ready for the answers.

Okay, everyone, well done.

Let's go for the answers for question one.

We can use what we've learned to work out where the centre of enlargement is for each transformation.

And by doing that, we can see that both A to A prime and C to C prime are both have the same centre of enlargement, whereas B two B prime, that centre of enlargement is somewhere different.

Therefore, these are our answers, A to A prime and C two C prime.

In question two, we have to describe these transformations fully.

We need to, first, say that it's in enlargement.

And we need to say it's by a scale factor of something and from the point somewhere.

The scale factor, we can work out by comparing corresponding lengths.

In this case, we can see four divided by two gives a scale factor of two.

And to get the centre of enlargement, we can do it by drawing lines, see where they intersect.

And in this case, we can see the centre of enlargement is at point (2,3).

So the full description is object A has been enlarged by a scale factor of two from the point (2,3).

Did you say enlarged? Did you say the scale factor? And did you say the coordinate of (2,3)? And in B, we need to do the same.

Object B has been enlarged by a scale factor of? We can get that by doing one divided by three, which is one third.

And from the point, we can get that by drawing our lines and see it's at a point (10,4).

So object B have been enlarged by a scale factor of a third from the point (10,4).

Great job today.

Let's summarise what we've learned in today's lesson.

During an enlargement, the scale factor is the multiplier that describes how large the image is compared to the object.

The centre of enlargement defines where the image will be placed in relation to the object.

And the centre of enlargement can be found by drawing ray lines through the corresponding vertices of the object and the image and seeing where they intersect.

Describing an enlargement fully requires both the scale factor and the centre of enlargement.

And the object and the enlarged image have the same orientation for positive scale factors.