Enlarging shapes
The chapter explores shape enlargements. It covers enlarging shapes using scale factors and centres of enlargement. And then goes on to cover enlarging shapes within a coordinate grid and then describing enlargements.
When working with shape enlargements you always need to indicate how many times bigger the shape has been made from the original shape. This is known as scale factor, for example observe the enlargement below;
[IMAGE]
The shape above has been enlarged by a scale factor of two from the original. On the grid we can also see the centre of enlargement at (-7, 5). It can be difficult to enlarge shapes that are not drawn on the grid. We can use projection lines to help us in the enlargement. Below the red shape is being enlarged to give the blue shape. The scale factor in this example is 2 and the centre of enlargement is at x.
[IMAGE]
Enlarging scale factor 2
Below we shall do an enlargement of the shape by a scale factor of 2 from the centre of enlargement.
[IMAGE]
First we draw a projection line from the centre to one of the corners of the shape, and onwards.
[IMAGE]
Then we measure the distance from the centre to the corner.
[IMAGE]
Since we want to enlarge the shape by a scale factor of 2 we want to double the distance to the point.
[IMAGE]
Above we’ve drawn a cross at the double-distance. We repeat this for the corners as well.
[IMAGE]
And lastly we just join up the crosses we’ve made.
[IMAGE]
The new shape we’ve created is a scale factor 2 enlargement of the small shape. You can do this for any enlargement scale factor given such as 3, ½, 100, etc. All you have to do is multiply with the original values given such as the lengths.
Scale Factor 3
Another example to show a different scale factor. The same steps apply.
[IMAGE]
First we drew a line from the centre point that was provided passing through the corners of the shape to enlarge. Then we measured the distance from the centre to the corners. Then we simply multiply this by three since the enlargement scale factor given is a 3. We measure the new distance from the enlargement centre and plot the points. Then we join up the points resulting in a new shape as shown below;
[IMAGE]
The green shape is a scale factor 3 enlargement of the red shape.
Negative scale factor enlargement
The principle applies here. Negative scale factors rotate the enlarged shape and put it on the opposite side. Below a scale factor -2 enlargement has been carried out. Below is a shape and the centre of enlargement to be used.
[IMAGE]
First we draw a projection line from the corners of the shape passing through the centre of enlargement as shown below;
[IMAGE]
We measure the distance from the corner to the centre then double it, above we can see it is 2.2cm so that doubles to 4.4. We measure it on the projection line and do a cross as shown below.
[IMAGE]
We repeat this step for all corners.
[IMAGE]
And lastly join up the points to create the new shape.
[IMAGE]
The blue shape is a scale factor -2 enlargement of the red shape.
