Exercise 6 Page 336

We want to decide which figures represent a dilation with a scale factor of 12.

Before we begin, recall that dilation is a transformation that enlarges or reduces the original figure proportionally. There are two types of dilation.

1. Enlargement: The image is larger than the preimage and is produced by a scale factor greater than 1.
2. Reduction: The image is smaller than the preimage and is produced by a scale factor between 0 and 1.Our scale factor is between 0 and 1. This means that the dilation is a reduction. 0 < 1/2 < 1 Now, the scale factor is the ratio of a length in the image to the corresponding length in the preimage. The ratio of 12 tells us that each length in the image is twice as short, or half as long, as the corresponding length in the preimage. Length in the Image/Length in the Preimage = 1/2 Let's take a look at our graph. We will be looking for pairs of figures where one has sides half as long as the other.
   

   The first potential pair is Figures 2 and 1. We see that the base of Figure 1 is 1 unit, and the base of Figure 2 is 2 units.

   

   The base of Figure 1 is exactly half the base of Figure 2. In the graph, we see that this is also true for the other pairs of lengths. This tells that Figure 1 is a dilation of Figure 2 with a scale factor of 12.

   

   Similarly, the base of Figure 2 is half as long as the corresponding base of Figure 3.

   

   From this we know that Figure 2 is a dilation of Figure 3 with a scale factor of 24= 12.

   

   We can see that there are no other pairs of figures where one has sides twice as short, or half as long, as the other.