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5.11 Dilations

Lesson

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 $\triangle ABC$ABC has side lengths of $3$3 cm, $4$4 cm and $5$5 cm. If it is dilated by a scale factor of $2$2 to produce $\triangle XYZ$XYZ, then $\triangle XYZ$XYZ will have side lengths of $6$6 cm, $8$8 cm and $10$10 cm, as shown below.

$\frac{6}{3}$63 $=$= $\frac{8}{4}$84 $=$= $\frac{10}{5}$105 $=$= $2$2

 

If any of these ratios were not equal to $2$2, then this would not be a dilation.

Dilating a shape

Well, we need two things:

  1. 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.
  2. 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. 
Remember!

A scale factor can increase or decrease the size of the new shape e.g. A scale factor of $3$3 means the new shape will have side lengths $3$3 times as big, whereas a scale factor of $\frac{1}{2}$12 means the new shape will be $\frac{1}{2}$12 as big as the original.

In general,

  • If the scale factor, $k$k, has $k>1$k>1, the image will be larger than the preimage
  • If the scale factor, $k$k, has $00<k<1, the image will be smaller than the preimage

 

Dilations on the coordinate plane

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$3 from the origin. Let's have a look at the coordinates of the vertices of the rectangles.

Dilation by a scale factor of $3$3
Center of dilation $\left(0,0\right)$(0,0)

Preimage Image
$A$A $\left(1,2\right)$(1,2) $A'$A $\left(3,6\right)$(3,6)
$B$B $\left(1,0\right)$(1,0) $B'$B $\left(3,0\right)$(3,0)
$C$C $\left(2,0\right)$(2,0) $C'$C $\left(6,0\right)$(6,0)
$D$D $\left(2,2\right)$(2,2) $D'$D $\left(6,6\right)$(6,6)

What do we notice about both the $x$x and the $y$y coordinates of the preimage and image?

Dilations

With a scale factor of $k$k and a center of dilation $\left(0,0\right)$(0,0), the preimage point $\left(x,y\right)$(x,y) will become the image point of $\left(kx,ky\right)$(kx,ky).

The ratios of corresponding side lengths must be the same, for example in a rectangle:

$\frac{A'B'}{AB}=\frac{B'C'}{BC}=\frac{C'D'}{CD}=\frac{D'A'}{DA}$ABAB=BCBC=CDCD=DADA

 

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 $\left(0,0\right)$(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?

 

Dilations using rays

We can enlarge shapes even without a coordinate plane. We just need a ruler.

1. Draw a point outside the shape. This will be your center of dilation.

2. Using a ruler, measure from the center of dilation to each of the vertices in your shape and record the distances.

3. Multiply and record all the distances you found in step 3 by the scale factor. Our dilation in this example is 3.

4. Draw each of the lines from the center of dilation, through the corresponding side of the existing shape, to the length you calculate in step 4.

5. Join up the points at the ends of the new lines you have draw and there you have it- your new shape!

 

Worked example 

Question 1

A rectangle with vertices $A$A$\left(-8,8\right)$(8,8), $B$B$\left(8,8\right)$(8,8), $C$C$\left(8,-8\right)$(8,8) and $D$D$\left(-8,-8\right)$(8,8) is dilated using the origin as the center of dilation. The vertices of the new rectangle are $A'$A$\left(-4,4\right)$(4,4), $B'$B$\left(4,4\right)$(4,4), $C'$C$\left(4,-4\right)$(4,4) and $D'$D$\left(-4,-4\right)$(4,4). What is the scale factor?

Think: The original coordinate values would be multiplied by the scale factor to give the new coordinate values.

Do: The length of side $AB$AB is $16$16 units. The length of side $A'B'$AB is $8$8 units. $\frac{8}{16}=\frac{1}{2}$816=12, so the scale factor is $\frac{1}{2}$12.

 

Practice questions

Question 2

Identify if rectangle $A'B'C'D'$ABCD is a dilation of rectangle $ABCD$ABCD.

Loading Graph...

  1. no

    A

    yes

    B

    no

    A

    yes

    B

Question 3

Identify if quadrilateral $A'B'C'D'$ABCD is a dilation of quadrilateral $ABCD$ABCD.

Loading Graph...

  1. yes

    A

    no

    B

    yes

    A

    no

    B

Question 4

Dilate the figure by a factor of $\frac{1}{2}$12, using the origin as the center of dilation.

  1. Loading Graph...

 

Outcomes

8.7a

Given a polygon, apply transformations, to include translations, reflections, and dilations, in the coordinate plane

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