Before we begin, recall that a dilation is a transformation that enlarges or reduces the original figure proportionally. There are two types of dilation.
Enlargement: The image is larger than the original figure, and is produced by a scale factor greater than 1.
Reduction: The image is smaller than the original figure, and is produced by a scale factor less than 1.
We will determine the given dilation first. Then we can find the scale factor.
Dilation
Let's analyze the given dilation.
We can tell that the blue image is bigger than the black preimage. Therefore, the dilation is an enlargement.
Scale Factor
The scale factor is the ratio of a length on the image to a corresponding length on the preimage. We can find the lengths of the corresponding sides of our figures in the following diagram.
Now we can find the length of these sides using the Distance Formula.
Figure
Vertices
sqrt((x_2-x_1)^2+(y_2-y_1)^2)
Simplify
Preimage
(0,0), (0,2)
sqrt(( - )^2+(2- )^2)
2
Image
(0,0), (0,3)
sqrt(( - )^2+(3- )^2)
3
The distance between the vertices of the preimage is 2, and between the vertices of the image the length is 3.
Finally, we can find the scale factor.
3/2=3/2
The scale factor of our dilation is 32.
Alternative Solution
Measure using the grid
We can also find the lengths of the corresponding sides of our figures with the help of the following diagram.
Knowing this, we can find the scale factor.
3/2=3/2
The scale factor of our dilation is 32.
