In this video we're going to look at enlargements. An enlargement changes the size of a shape, but importantly, it keeps all of the lengths in proportion. Let's have a look at some examples.
So if we take this triangle here, we can see the base has a length of 2 squares, and the height is 3 squares. So if we were to double both of those, we would have enlarged the shape. You can now see the base has doubled from 2 to 4 squares, and the height has doubled from 3 to 6. So this is an enlargement.
or we could triple all of the lengths. So for this triangle here, the base is now 2 multiplied by 3, which is 6, and the height is 3 multiplied by 3, which is 9. So this is also an enlargement. To go from the smallest triangle to the middle triangle here, we would say that this has a scale factor 2. That's because all of the lengths have been multiplied by 2. To go from the small triangle to the largest triangle, we would say this is scale factor 3, since all of the lengths have been multiplied by 3. It's important to understand that just because a shape has changed size, it doesn't necessarily mean it's an enlargement.
It's important that all of the lengths are in proportion. For example, if we change this middle triangle here so the base is now 6, this is not an enlargement. The reason is the base has been multiplied by 3, since 2 times 3 is 6, but the height has been multiplied by 2. We need these numbers to be the same if it's going to be an enlargement.
So this is not an enlargement. We've said that an enlargement is when we change the size of a shape. We typically think of enlarging something as making it bigger, but you can actually make something smaller and it still be classed as an enlargement. So if we take this trapezium here, and then shrink it down, you can see the base on the original one was 8 squares, but the base is now 4 squares, so it's half the size. So we do call this an enlargement even though it's been made smaller, we just say that the scale factor was 1 half.
Or we could take this triangle here, and then shrink it down like this. And you can see for this one the base was originally six squares, and now it's two squares, so it's been divided by three. So it's one third of the size. So the scale factor for this one is one third. Let's have a look at how some exam questions are worded.
So they might give you a shape on a grid like this, and say on the grid draw an enlargement of shape A with scale factor 3. So if we look at the width of the shape here we can see it's two squares, and since we need to do scale factor 3 we multiply this 2 by 3. which would give you 6. So the width of the enlarged shape will be 6. So let's draw a line that's 6 squares long. Then if we look at the height of the shape, we can see it's 3 squares, so we multiply this by 3 as well, which is 9. So we draw a line down from here, which is 9 squares long. It's now quite easy to finish this shape and join it up to make a rectangle.
For a second example, let's take this triangle here, and it says on the grid, draw an enlargement of shape B with scale factor 2. So if we look at the width of this shape here we can see it's four squares and since we're doing scale factor two it needs to get twice as long and four times two is eight. So the length of the top of the triangle there will be eight squares long. Then the height of the triangle is three squares so we need to double this to make six squares.
So I'm going to draw this dotted line down the middle of the triangle here which is six squares long and then connect those up to the end points to make the enlarged triangle. Now let's try another example. So this time we're actually going to do a different scale factor, we're going to do scale factor one half.
This means all of the lengths need to be half as big. So this one here at the top is two squares and since it's scale factor half we need to half two squares which is one square. So we just need to draw one line that's one square. Then this one here is also two squares so that will become half of two which is one again. This one here is four squares and half of four is two so we go across two.
This one here is two and half of two is one so we go down one. This one all the way along the bottom is 6 and half of 6 is 3 so we go across 3 and it's probably quite easy to see how we finish this shape now. This one's 4 and half of 4 is 2. So this will be a shape that's enlarged by scale factor 1 half.
All of the lengths are half as long. And let's try one more. So we've got this trapezium here, shape B, and it says to draw an enlargement of this shape with scale factor 1 third.
So if we look at the base of this one we count 9 squares and since it's scale factor 1 third we need to do a third of this. 1 third of 9 is the same as 9 divided by 3, which is 3, so the base will be 3 squares long. Then we've also got this length here on the top which is 3 squares, and we also have the height at being 3 squares and 1 third of 3 is 1. So we need to draw a length of 1 on the top, at a height of 1, so the trapezium will look like this.
If I take this rectangle here and enlarge it by scale factor 2, I could draw the answer like this, but there's no reason why I drew it in this particular place. I could have drawn it up here on the grid instead. or I could have drawn it over here, or I could have even drawn it down here.
In fact it would still be okay to draw it overlapping the shape somewhere like this. All of these are examples of a scale factor 2 enlargement. Sometimes exam questions do want you to draw the answer in a particular place though. Let's have a look at one of those. So in this question here it says enlarge the rectangle with scale factor 2 from the point P, and they'll mark on a point P onto the diagram like this.
So now we do need to draw that enlargement but the point P will tell us where to draw the final answer. Point P has a special name, it's known as the centre of enlargement. To start we're going to mark on a point on the rectangle.
I'm going to go for the one in the top left here. We're then going to consider a journey getting from P, the centre of enlargement, to that point I just marked. So to get from P to that point I just marked I could go across 1, 2, 3, 4 squares and then go down 1 square.
When doing an enlargement we repeat this journey again. so that the total number of times the journey's been taken is equal to the scale factor in the question. So the scale factor in this question was 2, so we need to do this journey from P to the black cross two times in total. We've already done it here once, so we need to do it once more. So the journey was four squares to the right and one square down, so from the black cross I'm going to go four squares to the right, 1, 2, 3, 4 and one square down, so the black cross would end up here.
We then do this same process for all of the other corners of the shape. So next I'm going to select this one here in the top right. So I start at the point P, the centre of enlargement, and I work out a journey to get to this green cross.
So that's 1, 2, 3, 4, 5, 6, 7 to the right, and then go down one square. I need to do that once more since it's scale factor 2. So 1, 2, 3, 4, 5, 6, 7 to the right, and one down. So the green cross would end up here. Now let's do the bottom left. So I need to get from the centre of enlargement, the red one, to the blue cross.
So this time I could go one, two, three, four to the right, and one, two, three down. I need to repeat this one more time so that the journey's been done twice in total. One, two, three, four to the right, one, two, three, down.
So the blue cross goes here. Now we've got one more point to do, that's the one in the bottom right corner. So we go from the centre of enlargement P to this cross.
So we go 1, 2, 3, 4, 5, 6, 7 to the right, 1, 2, 3 down. So we repeat that once more. 1, 2, 3, 4, 5, 6, 7 to the right, 1, 2, 3 down. So this cross goes here. If you now join up these crosses, you make the enlarged rectangle.
You can see this is the same as the rectangle I drew earlier, but now it's in the correct place. Let's try a second example like this. So this time we're going to do this triangle here. but we're going to enlarge it with a scale factor of 3 from the point Q, so the point Q is here. So we're going to pick a point on the triangle, I'm going to go for this one in the bottom right, and I'm going to do a journey from Q to this cross.
This journey is quite easy, it's 1, 2 to the left. Now since this is scale factor 3 this time, I need to do that journey 3 times in total. So I've already done it once, 2 to the left, I'll do it a second time, 1, 2 to the left, and a third time, 1, 2 to the left. So the black cross would go here.
Let's do the point in the bottom left. So from Q to this green cross is 1, 2, 3 to the left. That's the journey done once.
I need to do it a second time. 1, 2, 3 to the left. And a third time.
1, 2, 3 to the left. So the green cross goes here. Now let's finish with the point at the top of the triangle.
So to get from Q to this blue cross, 1, 2, 3 to the left, and 1, 2 up. So I do that a second time. 1. 2, 3 to the left, 1, 2, up. And a third time, 1, 2, 3 to the left, 1, 2, up. And the blue cross goes here.
So if you join up these points we end up with the enlarged triangle, but it's in the correct place. There's a useful way of checking if you've drawn your shape correctly. If you draw a straight line from the center enlargement, so if we go from P to one of the points on the shape, but then continue this straight line, it should hit the same point on the enlarged shape, like this.
This will apply to all of the points on the shape. So if we go to this one in the bottom left, and continue it straight, it hits the bottom left. This one in the top left, hits the top left, and the one in the bottom right, hits the bottom right. If any of these are misaligned, it might indicate you've done the enlargement wrong. We can check this on the triangle as well, and you can see that these ones have been enlarged correctly.
Sometimes we have to do the same process in reverse, so for example, an enlargement may have already been done for you, and the question could say… Shape A is enlarged to give shape B. And for the first part we need to write down the scale factor of enlargement. So this is how many times bigger the lengths are on B than they are on A. If we look at the height of A we can see it's 4 squares tall, and if you do the height of B you can see it's 12 squares tall. 4 multiplied by 3 gives us 12, so the scale factor must be 3. For part B of the question it might ask you to mark across where the centre of enlargement is.
To do this we draw those lines on again but in reverse. So if we start at the top of shape B, and then connect it up to the top of shape A, like this with a straight line, but then continue, if we then do this with another point, so for example the bottom ones here, and then continue, you'll see that they both cross at the same point. This point here must be where the centre of enlargement was.
Sometimes exam questions will be drawn on a coordinate axis. For example we could be given the shape A like this, and asked to enlarge the shape A, with scale factor 2, from the centre of enlargement. The difference here is rather than marking on a point P, they've told us the centre of enlargement is 0, 0, so we need to mark that on ourselves.
So let's mark across at 0, 0 and enlarge the shape as we did previously. Let's pick a point on the shape, I'm going to go for the bottom left, and we'll do a journey from the centre of enlargement to that shape. So this one's quite a short journey, one to the right, one up.
Now since we're doing this one scale factor 2, we need to repeat this a total of two times. We've already done it once, so we'll go a second time, one to the right. one up. So the green cross goes here.
Let's now do the bottom right corner. So from the centre of enlargement, which is at 0, 0, we go 1, 2, 3 and up 1. So we need to go to the right, 1, 2, 3 and then up 1. So the blue cross goes here. Let's complete it with the top left corner.
This one is 1 to the right and 2 up. So we do that once more, 1 to the right and 2 up and it ends up here. So this is where the final triangle will go.
We could do a quick check with some straight lines from the centre of enlargement, and you can see that this one has been enlarged correctly. They could also do these questions in reverse again, so they may give you an enlargement that's already been done, and ask you the following. Describe fully the single transformation that maps shape B to shape C.
When it says the word single transformation here, it's referring to the four different transformations of shapes you need to know. They are rotation, reflection, translation and enlargement. Now in this video obviously it's an enlargement, but you do need to be aware of the other four. In the question it won't tell you which one it is, so you'll have to select that yourself. It's usually pretty obvious if it's an enlargement because the shape will have changed size.
So we actually get some credit and score a mark for writing down the word enlargement. Then we need to give the scale factor. So to do this, I'm going to look at some lengths of the shape.
I can see the height of B is 3 squares, and the height of C is 6 squares. 3 times 2 is 6, so the scale factor must be 2. And then finally we also need to give the centre of enlargement. To find the centre of enlargement we're going to draw straight lines through the common points on the shape again. So if I join up these points here and then also these two points in the bottom right corner here, you can see these two lines cross at the point. So we give the centre of enlargement as a coordinate if we can.
It's important that you give all three bits of information here to score full marks on this question. Let's try a few more examples like these. So in this question we have rectangle A and it says to enlarge shape A with scale factor 3 from the centre of enlargement negative 8, 3. So we've got the centre of enlargement negative 8, 3. We need to mark that on first.
That goes here. Then we'll pick a point on the shape. So I'm going to go for this one and we'll work out the journey from the centre of enlargement to that point. So this one is 1, 2 to the right and 1 up. We need to do this one scale factor 3 so we need to do that journey three times in total.
We've already done it once. So let's repeat it a second time. One, two to the right, one up. And a third time, one, two to the right, one up. So it ends up here.
Now let's do this point here. So the journey this time is one, two to the right, one down. That's the journey once. So we need to go one, two to the right, one down.
And the journey a third time, one, two to the right, one down. So it ends up here. Now let's do this point.
So from the center of enlargement to that point, one, two, three to the right, one up. A second time, one, two, three to the right, one up. And a third time, one, two, three to the right, one up. This point will go here. Now we could do the last point but you could probably tell where it's going to go because this shape must form a rectangle.
So that's this one here. So the rectangle looks like this. Let's draw some straight lines from the centre enlargement through points to see if it lines up.
So if we draw this line here through the blue points and this one here through the green ones you can see it looks like we've done this enlargement correctly. Let's try another one where we have to do a description. So we've got these two squares, B and C, and it says to describe fully the single transformation that maps shape B to shape C.
Now you've got to be really careful here, this is going from shape B to shape C. So the shape is getting smaller, but we know this is still an enlargement. So where the question says single transformation, we need to pick one of the four transformations.
You know it's an enlargement because that's what this video is about. But we do need to write that down to get a mark. Then we need the scale factor.
So if we look at the height of B, that's four squares, and the height of C, that's one square. So you may think initially this is scale factor 4, but remember we're going from B to C. So the shape has got smaller. In fact, it's one quarter of the size. So the scale factor will be one quarter.
Then we need the centre of enlargement. So let's pick a point on B and connect it up to the same point on C and continue that line and do another point like this and you can see these two lines cross at a common point which is seven zero. So the centre of enlargement has coordinates seven zero.
Let's try one more set of examples. So for this one we've got this shape A here and it says to enlarge shape A with scale factor one half this time from the centre of enlargement negative two seven. So we need to mark on this center of enlargement, negative 2, 7, that goes here.
And let's pick a point on the shape to start with. I'm going to go for this one here. So let's work out a journey from the center to this point. So it's 1, 2, 3, 4 to the right, and 1, 2, 3, 4 down. Now this time we're doing scale factor 1 half.
So rather than doing the journey 2 times, like for scale factor 2, or 3 times, like for scale factor 3, we need to do half of a journey. So this journey was 4 to the right and 4 down. half of that journey will be two to the right and two down. So let's remove this and go two to the right and two down.
So the green cross will actually go here this time. Now let's do this point here, so from the center to the blue cross, one two three four five six to the right and two down. We need to do half of this, so instead of six right, three right, instead of two down, one down. So we need to go 1, 2, 3 to the right and 1 down.
So the blue cross goes here. Now let's do the one in the top right. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 to the right and 2 down.
Half of this journey would be 5 to the right and 1 down. So let's go 1, 2, 3, 4, 5 to the right and 1 down. So this one goes here and you can probably see where the one in the bottom right needs to go since this shapes a parallelogram.
So if we join that up it ends up like this. And we can draw on some straight lines to check. So this time I'm going to do the bottom right corner and you can see that does hit the bottom right corner and also the top right corner and you can see that one hits too.
So this enlargement is in the correct place. Let's do one more question a bit like this. So we've got this shape B here and we're going to enlarge it with scale factor 1 third from the centre, 7, negative 5. So let's plot the centre first 7, negative 5, that goes here and let's pick a point on the shape, I'm going to start with this one.
So the journey this time is 1, 2, 3, 4, 5, 6, 7, 8, 9 to the left, but the scale factor is 1 third, so we need to do 1 third of 9, which is 9 divided by 3, which is 3. So we need to do 1, 2, 3 to the left, so the green one goes here. Now let's do this one over in the bottom left. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 15 to the left and we need to do one third of that which is five.
So we need to go five to the left. So let's go one, two, three, four, five to the left. The blue one goes here. There's one more point to do at the top here. We go one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve to the left and one, two, three, four, five, six up.
So we need to do one third of those. One third of twelve to the left 12 divided by 3 is 4 to the left, and one third of 6 up is 6 divided by 3, which is 2 up. So if we undo all of that, and we need to go 1, 2, 3, 4 to the left, and 1, 2 up. So this point goes here.
If we join this up we can make the triangle, and let's draw some straight lines to check, and that looks good to me. Thank you for watching this video, I hope you found it useful. Check out the one I think you should watch next.
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