### Dilation

Two figures are similar if they are the same shape but not necessarily the same size. One way to create similar figures is by dilating. A dilation makes a figure larger or smaller but the new resulting figure has the same shape as the original.

**Dilation:** An enlargement or reduction of a figure that preserves shape but not size. All dilations are similar to the original figure.

Dilations have a **center** and a **scale factor**. The center is the point of reference for the dilation and the scale factor tells us how much the figure stretches or shrinks. A scale factor is labeled . Only positive scale factors, k, will be considered in this text.

If the dilated image is smaller than the original, then .

If the dilated image is larger than the original, then .

A dilation, or image, is always followed by a .

Label It |
Say It |
---|---|

“prime” (copy of the original) | |

“a prime” (copy of point ) | |

“a double prime” (second copy) |

What if you enlarged or reduced a triangle without changing its shape? How could you find the scale factor by which the triangle was stretched or shrunk?

### Examples

#### Example 1

Find the perimeters of and . Compare this ratio to the scale factor.

The perimeter of . The perimeter of . The ratio is 80:40, which reduces to 2:1, which is the same as the scale factor.

#### Example 2

is a dilation of . If is the center of dilation, what is the scale factor?

Because is a dilation of , then . The scale factor is the ratio of the sides. Since is smaller than the original, , the scale factor is going to be less than one, .

If was the dilated image, the scale factor would have been .

#### Example 3

The center of dilation is and the scale factor is 3.

Find .

If the scale factor is 3 and is 6 units away from , then is going to be units away from . The dilated image will be on the same line as the original image and center.

#### Example 4

Using the picture above, change the scale factor to .

Find using this new scale factor.

The scale factor is , so is going to be units away from . will also be collinear with and center.

#### Example 5

is a rectangle. If the center of dilation is and , draw .

If is the center of dilation, then and will be the same point. From there, will be units above and will be 12 units to the right of .

### Review

For the given shapes, draw the dilation, given the scale factor and center.

- , center is

- , center is

- , center is

- , center is

In the four questions below, you are told the scale factor. Determine the dimensions of the dilation. In each diagram, the **black** figure is the original and is the center of dilation.

In the three questions below, find the scale factor, given the corresponding sides. In each diagram, the **black** figure is the original and is the center of dilation.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 7.11.