13.06 Dilations
We've learnt 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:
- A centre of dilation: a point from where we start the enlargement. This may be inside or outside the original shape.
- 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.
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 $0
Dilation using vertex coordinates
1. Find the distance from the centre of dilation to a point on the object.
2. Using the given scale factor, draw the line from the centre 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.
If a shape is being dilated by a factor of $a$a, then the following mapping occurs:
$(x,y)\to(ax,a)$(x,y)→(ax,a)
For example, If a shape was being dilated by a factor of $2$2 then $(x,y)\to(ax,a)$(x,y)→(ax,a), so each of the image points are double the original points. For example, the image point for $(2,-3)$(2,−3) would be $(4,-6)$(4,−6).
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 centre of dilation.
2. Using a ruler, measure from the centre 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 centre 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 centre 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 $A'B'$A′B′ 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 $A'B'C'D'$A′B′C′D′ is a dilation of rectangle $ABCD$ABCD.
no
Ayes
B
Question 3
Identify if quadrilateral $A'B'C'D'$A′B′C′D′ is a dilation of quadrilateral $ABCD$ABCD.
yes
Ano
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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