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4.3 J) Enlargement: Negative Scale Factors
4.3 J) Enlargement: Negative Scale Factors
A negative scale factor makes the shape appear on the other side of the centre of enlargement, which can make the shape look like it is upside down.
The process to answer a question with a negative scale factor is exactly the same as the process that we have looked at in the previous sections.
The process to answer a question with a negative scale factor is exactly the same as the process that we have looked at in the previous sections.
Example 1
Enlarge the shape below with a scale factor of -½ with the centre of enlargement at (0, 2). Label the points on the enlarged shape A’B’C’. Click here for a printable version of this question.
Enlarge the shape below with a scale factor of -½ with the centre of enlargement at (0, 2). Label the points on the enlarged shape A’B’C’. Click here for a printable version of this question.
Before we start answering the question, it is worth going through some observations about the question. The first thing to notice in this question is that we have a negative scale factor, which means that the enlarged shape will appear on the other side of the centre of enlargement. Also, the scale factor is a fraction (between 0 and -1), which means that the enlarged shape will be smaller than the original shape.
The first step in answering an enlargement question is to mark the centre of enlargement; we are told in the question that the centre of enlargement is (0, 2) and this is plotted on the diagram below.
The first step in answering an enlargement question is to mark the centre of enlargement; we are told in the question that the centre of enlargement is (0, 2) and this is plotted on the diagram below.
We have already established that the enlarged shape will appear on the other side of the centre of enlargement and this is because of the negative scale factor. Therefore, we draw lines going from the points on the original shape to and through the centre of enlargement. The working is shown below.
The next step is to work out the distance between the centre of enlargement and a point on the shape. We then multiply this distance by the scale factor, which will tell us the distance between the centre of enlargement and the respective point on the enlarged shape.
I am going to do this for point C and C’ first. The distance between the centre of enlargement and C is 4 to the right. We now multiply this distance by the scale factor. The scale factor for this question is -½, which is negative. A negative scale factor changes the direction of the distance; for example, a rightwards distance would now become leftwards, and an upwards distance would become downwards. The distance between the centre of enlargement and C was 4 to the right. We multiply this distance by the scale factor (-½), which results in the direction changing because of the negative (the distance will be towards the left rather than the right) and the distance halving (the distance will be 2 rather than 4). Therefore, the distance between the centre of enlargement and C’ is 2 to the left. Point C’ is shown on the diagram below.
I am going to do this for point C and C’ first. The distance between the centre of enlargement and C is 4 to the right. We now multiply this distance by the scale factor. The scale factor for this question is -½, which is negative. A negative scale factor changes the direction of the distance; for example, a rightwards distance would now become leftwards, and an upwards distance would become downwards. The distance between the centre of enlargement and C was 4 to the right. We multiply this distance by the scale factor (-½), which results in the direction changing because of the negative (the distance will be towards the left rather than the right) and the distance halving (the distance will be 2 rather than 4). Therefore, the distance between the centre of enlargement and C’ is 2 to the left. Point C’ is shown on the diagram below.
We now do the same for the other two points:
We now plot the points and draw lines to create our enlarged shape.
- The distance between the centre of enlargement and A is 2 to the right and 4 up. We multiply this distance by -½ to obtain the distance between the centre of enlargement and A’, which results in the distance being 1 to the left and 2 down.
- The distance between the centre of enlargement and B is 4 to the right and 4 up. We multiply this distance by -½ to obtain the distance between the centre of enlargement and B’, which results in the distance being 2 to the left and 2 down.
We now plot the points and draw lines to create our enlarged shape.
From the above diagram we can see that when we have a negative scale factor, the shape appears upside down.
Example 2
We are now going to have an example whereby we find the scale factor.
The shape ABCDEF has been enlarged to create the shape A’B’C’D’E’F’. Describe the enlargement.
We are now going to have an example whereby we find the scale factor.
The shape ABCDEF has been enlarged to create the shape A’B’C’D’E’F’. Describe the enlargement.
The first thing that we notice about this enlargement is that the shape after the enlargement appears to be upside down. This means that we are going to have a negative scale factor. This does not change the way that we work out the scale factor; it just means that we add a negative to our scale factor. We work out the scale factor by dividing the length of one of the sides on the enlarged shape by the length of the respective side on the original shape. I am going to use the lengths of A’B’ and AB; A’B’ is 2 units and AB is 1 unit. We complete the following calculation to find the scale factor:
We now add the negative, which means that the scale factor is -2. We could check that we have obtained the correct scale factor by using another side on the shape (I am not going to do this, but feel free to give it ago).
The next step is to find the centre of enlargement and we do this by drawing lines connecting the points on the original shape to their respective points on the enlarged shape; we draw lines going from A to A’, B to B’, …, F to F’.
The next step is to find the centre of enlargement and we do this by drawing lines connecting the points on the original shape to their respective points on the enlarged shape; we draw lines going from A to A’, B to B’, …, F to F’.
The centre of enlargement is the point where all of the lines intercept one another. We can see on the diagram that all of the lines intercept one another at the point with the coordinates (-2, 0).
Therefore, the answer to this question is an enlargement of a scale factor of -2 with a centre of enlargement at (-2, 0).
Therefore, the answer to this question is an enlargement of a scale factor of -2 with a centre of enlargement at (-2, 0).