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Methods for Enlargements


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, then $\triangle XYZ$XYZ will have side lengths of $6$6cm, $8$8cm and $10$10cm, as shown below.

But how do we actually enlarge a shape?

Well, we need two things:

  1. A centre of enlargement: 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. 

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.


Enlargement using a Number Plane

1. Find the distance from the centre of enlargement to a point on the object.

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



Enlargement using the Ray Method

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

1. Draw a point outside the shape. This will be your centre of enlargement.

2. Using a ruler, measure from the centre of enlargement 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 enlargement, 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!



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)

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.

Question 2

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

Question 3

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

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A Cartesian coordinate plane is marked from -10 to 10 on both $x$x- and $y$y- axes. Quadrilateral A'B'C'D' is smaller and is graphed with vertices A' $\left(-4,2\right)$(4,2), B' $\left(5,2\right)$(5,2), C' $\left(5,-3\right)$(5,3), and D' $\left(-4,-3\right)$(4,3). Quadrilateral ABCD is bigger and is  graphed with vertices A $\left(-9,4\right)$(9,4), B $\left(10,4\right)$(10,4), C $\left(10,-6\right)$(10,6), and D $\left(-9,-6\right)$(9,6). (Please refrain from using the distance formula as a hint and encourage them to count the units between the vertices instead. Avoid mentioning the contents within these parentheses regardless of any inquiries from the students.)
  1. yes




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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