9.02 Dilations
Dilations on the coordinate plane
We've learned that  similar polygons have all corresponding sides in the same ratio. So if a shape is enlarged or reduced, all the side lengths will increase or decrease in the same ratio. This enlargement or reduction is called a dilation. For example, let's say \triangle {ABC} has side lengths of 3\text{ cm}, 4\text{ cm}, and 5\text{ cm}. If it is dilated by a scale factor of 2 to produce \triangle {XYZ}, then \triangle {XYZ} will have side lengths of 6\text{ cm}, 8\text{ cm}, and 10\text{ cm}, as shown below.
\dfrac63=\dfrac84=\dfrac{10}5=2
If any of these ratios were not equal to 2, then this would not be a dilation.
Well, we need two things:
A center of dilation: a point from where we start the enlargement. This may be inside or outside the original shape, and for now we will only use the origin.
A scale factor: the ratio by which we increase/ decrease the shape. We calculate a scale factor just like we would calculate the ratio of the sides in similar triangles.
A scale factor can increase or decrease the size of the new shape e.g. A scale factor of 3 means the new shape will have side lengths 3 times as big, whereas a scale factor of \dfrac12 means the new shape will be \dfrac12 as big as the original.
In general,
If the scale factor, k, has k>1, the image will be larger than the preimage
If the scale factor, k, has 0<k<1, the image will be smaller than the preimage
We can use the coordinates of the vertices of polygons on the coordinate plane to find the image. Consider the image below which dilates the green preimage by a scale factor of 3 from the origin. Let's have a look at the coordinates of the vertices of the rectangles.
The dilation is by a scale factor of 3, and the center of dilation is at (0,0).
| Preimage | Image |
|---|---|
| A(1,2) | A'(3,6) |
| B(1,0) | B'(3,0) |
| C(2,0) | C'(6,0) |
| D(2,2) | D'(6,2) |
What do we notice about both the x and the y coordinates of the preimage and image?
With a scale factor of k and a center of dilation (0,0), the preimage point (x,y) will become the image point of (kx,ky).
The ratios of corresponding side lengths must be the same, for example in a rectangle:
\frac{A'B'}{AB}=\frac{B'C'}{BC}=\frac{C'D'}{CD}=\frac{D'A'}{DA}
Exploration
The applet below allows us to see the image for a rectangle we choose along with the scale factor of our choice.
Use the slider to select the desired scale factor and the three blue points on the preimage rectangle to make your preimage.
Can you make a preimage where the center of dilation (0,0) is inside the image? outside the image? on an edge of the image?
Can you make a preimage which is completely inside the image?
What scale factors make the image larger than the preimage? Which make is smaller? Which make it the same size?
A preimage where the center of dilation (0,0) can be inside the image, outside the image, on the edge of the image, and inside the image.
If the scale factor is more than 1, the image is larger than the preimage.
If the scale factor is less than 1, the image is smaller than the preimage.
If the scale factor is 1, the image has the same size as preimage.
Examples
Example 1
A rectangle with vertices A\left(-8,8\right), B\left(8,8\right), C\left(8,-8\right), and D\left(-8,-8\right) is dilated using the origin as the center of dilation. The vertices of the new rectangle are A'\left(-4,4\right), B'\left(4,4\right), C'\left(4,-4\right), and D'\left(-4,-4\right). What is the scale factor?
Example 2
Identify if rectangle A'B'C'D' is a dilation of rectangle ABCD.
Factors for shape dilation:
A center of dilation: a point from where we start the enlargement.
A scale factor: the ratio by which we increase/ decrease the shape.
A scale factor can increase or decrease the size of the new shape. In general:
If the scale factor, k, has k>1, the image will be larger than the preimage
If the scale factor, k, has 0<k<1, the image will be smaller than the preimage
With a scale factor of k and a center of dilation (0,0), the preimage point (x,y) will become the image point of (kx,ky).
The ratios of corresponding side lengths must be the same, for example in a rectangle:
\frac{A'B'}{AB}=\frac{B'C'}{BC}=\frac{C'D'}{CD}=\frac{D'A'}{DA}