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4.3 G) Enlargements: Normal – Part 1
4.3 G) Enlargements: Normal – Part 1
An enlargement changes the size of a shape; they can make shapes larger or smaller.
There are two pieces of information that we need to know in order to carry out an enlargement:
Let’s have some examples. Click here for a printable version of the examples.
There are two pieces of information that we need to know in order to carry out an enlargement:
- The scale factors, which tells us how much larger or smaller the enlarged shape is. A scale factor of 2 would make the lengths of the enlarged shape twice as long as the smaller shape.
- The centre of enlargement, which is the point that we are enlarging the shape from.
Let’s have some examples. Click here for a printable version of the examples.
Example 1
Enlarge the triangle (ABC) below with a scale factor of 2 with the centre of enlargement (-3, -2).
Enlarge the triangle (ABC) below with a scale factor of 2 with the centre of enlargement (-3, -2).
The first step in answering an enlargement question is to mark the centre of enlargement on the graph. We are told in the question that the centre of enlargement is (-3, -1). The x coordinates in the coordinates comes first and the y coordinate in the coordinates comes second (use along the corridor and up the stairs to help you remember). Therefore, the x coordinate is -3 and the y coordinate is -1. The centre of enlargement is marked on the graph below:
The next step is to draw lines going from the centre of enlargement through each of the points on the shape that we are enlarging. It is best to draw these lines quite faint as you do not want your diagram to become really confusing.
When we enlarge a shape by a scale factor from a centre of enlargement, the distances from the centre of enlargement to each point are multiplied by the scale factor. I am going to let the points for the enlarged shape be A’B’C’ and I am going to find the position of A’ first. It is best to split the distances between the point and the centre of enlargement in a horizontal and vertical component. Currently, point A compared to the centre of enlargement is 1 to the right and 2 up. The scale factor in this question is 2. This means that the distance between the centre of enlargement and point A’ will be double the distance between the centre of enlargement and point A. Therefore, compared to the centre of enlargement, point A’ will be 2 to the right (distance x scale factor; 1 x 2) and 4 up (2 x 2). We can see that this point lies on the faint construction line that we drew earlier, which is what should happened (it is a warning sign that you have done something wrong if the point that you have found does not lie on the construction line that you have drawn).
We now do the same for the other two points (point B/B’ and point C/C’).
The distance of point B from the centre of enlargement is 1 to the right and 1 up. We multiply these distances by the scale factor, which is 2. This means that compared to the centre of enlargement, point B’ will be 2 to the right and 2 up.
The distance of point C from the centre of enlargement is 3 to the right and 1 up. We multiply these distances by the scale factor (2), which tells us that compared to the centre of enlargement, point C’ is 6 to the right and 2 up.
We now plot the points and join the coordinates together to create the enlarged triangle A’B’C’.
The distance of point B from the centre of enlargement is 1 to the right and 1 up. We multiply these distances by the scale factor, which is 2. This means that compared to the centre of enlargement, point B’ will be 2 to the right and 2 up.
The distance of point C from the centre of enlargement is 3 to the right and 1 up. We multiply these distances by the scale factor (2), which tells us that compared to the centre of enlargement, point C’ is 6 to the right and 2 up.
We now plot the points and join the coordinates together to create the enlarged triangle A’B’C’.
From the shape above, we can see that the lengths of the sides on the new shape are twice the length of the sides on the shape that we were enlarging by a scale factor of 2. The length of AB is 1 unit and the length of A’B’ is 2 units. Also, the length of BC is 2 units and the length of B’C’ is 4 units.
Example 2
The centre of enlargement can be outside, inside or on the corner of a shape. The location of the centre of enlargement does not change the working. The next example will be an enlargement question where the centre of enlargement is inside the shape.
Enlarge the shape below with a scale factor of 3 with the centre of enlargement (2, 1).
The centre of enlargement can be outside, inside or on the corner of a shape. The location of the centre of enlargement does not change the working. The next example will be an enlargement question where the centre of enlargement is inside the shape.
Enlarge the shape below with a scale factor of 3 with the centre of enlargement (2, 1).
The first step in answering this question is to plot the centre of enlargement on the graph; the coordinates of the centre of enlargement are (2, 1) and I have plotted this point on the graph below.
The next step is to draw lines going from the centre of enlargement through each of the points on the shape that we are enlarging.
We now find the distances between the points on the shape and the centre of enlargement. I am going to start with point A. The distance between the centre of enlargement and point A is 1 to the right and 2 up. We now multiply the distances by the scale factor, and we are told in the question that the scale factor is 3. This means that the distance between the centre of enlargement and point A’ will be 3 to the right (distance x scale factor; 1 x 3) and 6 up (2 x 3). Point A’ is plotted on the graph below:
We now do the same for the other 3 points (B’, C’ and D’).
We plot these points and connect them to create the enlarged shape (A’B’C’D’).
- The distance between the centre of enlargement and B is 1 to the right and 2 down. This means that the distance between the centre of enlargement and B’ will be 3 to the right and 6 down.
- The distance between the centre of enlargement and C is 1 to the left. This means that the distance between the centre of enlargement and C’ will be 3 to the left.
- The distance between the centre of enlargement and D is 1 to the left and 1 up. This means that the distance between the centre of enlargement and D’ will be 3 to the left and 3 up.
We plot these points and connect them to create the enlarged shape (A’B’C’D’).