3.8.1 Dilations and Scale Factor

The student is able to (I can):

• Identify and draw dilations

• Identify scale factors and use scale factors to solve problems

dilation A transformation that changes the size of a figure but not the shape.

Example:

Tell whether each transformation appears to be a dilation.

1. 2.

SSyes no

scale factor The ratio of the image to the preimage.

If k < 1, the figure gets smaller; if k > 1, the figure gets larger.

•

•

•

X´

Y´

Z´•

•

•

P

X

Y

Z

center of dilation

X Y Y Z X Zk

XY YZ XZ

′ ′ ′ ′ ′ ′= = =

Example 1. What is the scale factor of the dilation?

2. If you are enlarging a 4x6 photo by a scale factor of 4, what are the new dimensions?

4(4) = 16 6(4) = 24

New dimensions = 16x24

10

24

5

12

5 1 12 1k (or k )

10 2 24 2= = = =

Scale factor and coordinates:

What point is the image of A under the dilation with the given scale factor with the center of dilation at 0?

1. k = 2

2(2) = 4, thus point D

2. k = −1

2(−1) = −2, thus point B

3. k =

| || ||| | ||0 2 4-2-4

••• •

AB C D

1

2−

12 1, thus point C

2

− = −

If P(x, y) is a point being dilated centered at the origin, with a scale factor of k, then the image of the point is P´(kx, ky).

Example: What are the coordinates of a triangle with vertices S(−3, 2), K(0, 4), and Y(2, −3) under a dilation with a scale factor of 3, centered at the origin?

S´(3(−3), 3(2)) = S´(−9, 6)

K´(3(0), 3(4)) = K´(0, 12)

Y´(3(2), 3(−3)) = Y´(6, −9)

Note: If k is negative, the resulting dilation will be rotated 180º about the center.

Examples Dilate the following vertices by the given scale factor. All dilations are centered about the origin.

1. B(2, 0), I(3, 3), G(5, −1); k=2

B´(4, 0), I´(6, 6), G´(10, −2)

2. T(-3, -3), I(-3, 3), N(6, 3), Y(6, -3); k=

T´(-1, -1), I´(-1, 1), N´(2, 1), Y´(2, -1)

3. S(−4, 2), E(−6, 0), A(−2, −4); k=

S´(2, −1), E´(3, 0), A´(1, 2)

1

3

1

2−