Learn on PengiBig Ideas Math, Course 3Chapter 2: Transformations

Lesson 7: Dilations

In this Grade 8 lesson from Big Ideas Math Course 3, students learn how to identify and perform dilations in the coordinate plane by multiplying vertex coordinates by a scale factor k to produce enlargements (k > 1) or reductions (0 < k < 1). Students practice locating the center of dilation, calculating scale factors, and distinguishing dilations from other transformations such as translations. The lesson supports Common Core standards 8.G.3 and 8.G.4 and builds toward using multiple transformations together to find images of figures.

Section 1

Locating the Center of Dilation

Property

If you are looking at a pre-image and its dilated image, you can work backwards to find the exact Center of Dilation. Because dilations expand outward in straight lines, drawing straight lines through corresponding vertices (connecting A to A', B to B', C to C', and extending them) will eventually make all the lines intersect at one single point. That intersection is the Center of Dilation.

Examples

  • Finding the Center: You have a small square PQRS and a large square P'Q'R'S'. Place a ruler on point P and point P', draw a long line. Do the same for Q and Q'. The exact spot on the graph where those two lines cross each other is your center of dilation.

Explanation

Think of the Center of Dilation as a flashlight, and the shape as an object casting a shadow. The light rays travel in perfectly straight lines through the corners of the object to create the enlarged shadow. By tracing the lines backwards from the shadow (image) through the object (pre-image), you will always find the flashlight (center). If you draw the lines and they are perfectly parallel and never cross, then the shape wasn't dilated—it was translated!

Section 2

Scale Factors

Property

The scale factor is the ratio of the lengths in the new figure to the corresponding lengths in the original figure.

scale factor=new lengthcorresponding original length \text{scale factor} = \frac{\text{new length}}{\text{corresponding original length}}
new length=scale factor×corresponding original length \text{new length} = \text{scale factor} \times \text{corresponding original length}

Examples

  • A map has a scale factor of 110000\frac{1}{10000}. A road that is 3 cm long on the map is 3×10000=300003 \times 10000 = 30000 cm, or 300 meters, in real life.
  • A triangle with a base of 8 inches is enlarged to a similar triangle with a base of 20 inches. The scale factor is 208=2.5\frac{20}{8} = 2.5.
  • To reduce a 12-foot wall to fit on a blueprint with a scale factor of 148\frac{1}{48}, its length on the blueprint is 12×148=1412 \times \frac{1}{48} = \frac{1}{4} foot, or 3 inches.

Explanation

The scale factor is the number you multiply by to change the size of a figure. A scale factor greater than 1 makes the figure bigger (an enlargement), while a factor between 0 and 1 makes it smaller (a reduction).

Section 3

Defining the Scale Factor (k): Enlargements and Reductions

Property

For a dilation with scale factor kk:

  • If k>1k > 1, the dilation is an enlargement (image is larger than original)
  • If 0<k<10 < k < 1, the dilation is a reduction (image is smaller than original)
  • If k=1k = 1, the image is congruent to the original figure

Examples

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Locating the Center of Dilation

Property

If you are looking at a pre-image and its dilated image, you can work backwards to find the exact Center of Dilation. Because dilations expand outward in straight lines, drawing straight lines through corresponding vertices (connecting A to A', B to B', C to C', and extending them) will eventually make all the lines intersect at one single point. That intersection is the Center of Dilation.

Examples

  • Finding the Center: You have a small square PQRS and a large square P'Q'R'S'. Place a ruler on point P and point P', draw a long line. Do the same for Q and Q'. The exact spot on the graph where those two lines cross each other is your center of dilation.

Explanation

Think of the Center of Dilation as a flashlight, and the shape as an object casting a shadow. The light rays travel in perfectly straight lines through the corners of the object to create the enlarged shadow. By tracing the lines backwards from the shadow (image) through the object (pre-image), you will always find the flashlight (center). If you draw the lines and they are perfectly parallel and never cross, then the shape wasn't dilated—it was translated!

Section 2

Scale Factors

Property

The scale factor is the ratio of the lengths in the new figure to the corresponding lengths in the original figure.

scale factor=new lengthcorresponding original length \text{scale factor} = \frac{\text{new length}}{\text{corresponding original length}}
new length=scale factor×corresponding original length \text{new length} = \text{scale factor} \times \text{corresponding original length}

Examples

  • A map has a scale factor of 110000\frac{1}{10000}. A road that is 3 cm long on the map is 3×10000=300003 \times 10000 = 30000 cm, or 300 meters, in real life.
  • A triangle with a base of 8 inches is enlarged to a similar triangle with a base of 20 inches. The scale factor is 208=2.5\frac{20}{8} = 2.5.
  • To reduce a 12-foot wall to fit on a blueprint with a scale factor of 148\frac{1}{48}, its length on the blueprint is 12×148=1412 \times \frac{1}{48} = \frac{1}{4} foot, or 3 inches.

Explanation

The scale factor is the number you multiply by to change the size of a figure. A scale factor greater than 1 makes the figure bigger (an enlargement), while a factor between 0 and 1 makes it smaller (a reduction).

Section 3

Defining the Scale Factor (k): Enlargements and Reductions

Property

For a dilation with scale factor kk:

  • If k>1k > 1, the dilation is an enlargement (image is larger than original)
  • If 0<k<10 < k < 1, the dilation is a reduction (image is smaller than original)
  • If k=1k = 1, the image is congruent to the original figure

Examples