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4.04 Dilations

Lesson

We've learned that similar triangles 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$3cm, $4$4cm and $5$5cm. 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$6cm, $8$8cm and $10$10cm, as shown below.

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.
  2. A dilation factor: the ratio by which we increase/ decrease the shape. We calculate a dilation factor just like we would calculate the ratio of the sides in similar triangles. 
Remember!

A dilation factor can increase or decrease the size of the new shape e.g. A dilation factor of $3$3 means the new shape will be $3$3 times as big, whereas a dilation 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 dilation factor, $k$k, has $k>1$k>1, the image will be larger than the preimage
  • If the dilation factor, $k$k, has $00<k<1, the image will be smaller than the preimage

 

Dilation using vertex coordinates

1. Find the distance from the center of dilation to a point on the object.

2. Using the given scale factor, draw the line from the center of dilation, through the original vertex until you reach the necessary distance. In our example, the dilation factor is $2$2, so instead of $2$2 units, our new line is going to be $4$4 units.

3. Repeat steps 1 and 2 for each point in the object.

4. Join up the points with lines to draw the image.

 

 

Dilation using rays

We can enlarge enlarge or reduce 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 dilation 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 dilation factor?

Think: The original coordinate values would be multiplied by the dilation 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 dilation 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.

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  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.

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Outcomes

GEO-G.CO.2

Represent transformations as geometric functions that take points in the plane as inputs and give points as outputs. Compare transformations that preserve distance and angle measure to those that do not.

GEO-G.SRT.1

Verify experimentally the properties of dilations given by a center and a scale factor.

GEO-G.SRT.1a

Verify experimentally that dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.

GEO-G.SRT.1b

Verify experimentally that the dilation of a line segment is longer or shorter in the ratio given by the scale factor.

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