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Geometry Transformations

Enlargements (Dilations)

Enlargement, sometimes called scaling or dilation, is a kind of transformation that changes the size of an object. The image created is similar to the object.

 This diagram shows the blue shaded shape enlarged by a scale factor of 2

The scale factor is how many times larger than the object the image is. For every enlargement, a scale factor, k, must be specified:

length of side in image = length of side in object x scale factor

Despite the name enlargement, it includes making objects smaller because the shape can be bigger or smaller according to the scale factor, k.

 This diagram shows the blue shaded shape enlarged by a scale factor of 1/2

Hot tips:

 If k>1, then the image is larger than the object. If k = $\pm$1, then the image and the object is the same size. If 0

For any enlargement, there must be a point called the centre of enlargement:

distance from centre of enlargement to point on image =
distance from centre of enlargement to point on object x scale factor

 This diagram shows the blue shaded triangle enlarged by a scale factor of 2. The centre of enlargement is (0,0) This diagram shows the blue shaded shape enlarged by a scale factor of 3. The centre of enlargement is (4,1)

With a negative scale factor the number part of it tells us if the shape is to be made bigger or smaller, while the negative sign tells us that the image and object are on opposite sides of the centre of enlargement.

 This diagram shows the blue shaded shape enlarged by a scale factor of -2. The centre of enlargement is (1,0)

Find the scale factor:

We know that k<1, because the image is smaller than the object.

$k=\frac{\bar{A\prime\;B\prime}}{\bar{AB}}=\frac{5}{10}=\frac{1}{2}$

Perform a dilation centered at (0,0), where k=3/2

Solution:

A(2,8) $\rightarrow$ A'(3,12)
B(10,4) $\rightarrow$ B'(15,6)

C(6,2) $\rightarrow$ C'(9,3)

Graphycally: