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Grade 912

7.01 Dilations

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Concept summary

A dilation is a transformation which changes the size of a figure through either an enlargement, which makes the figure bigger, or a reduction, which makes the figure smaller, by a given scale factor.

Dilation

A proportional increase or decrease in size in all directions.

There are two quadrilaterals of the same shape, with dashed arrows connecting the corresponding vertices.
Scale factor

The constant that is multiplied by the length of each side of a figure to produce an image that is the same shape as the original figure.

Enlargement

An increase in size without changing the shape. This corresponds to a dilation with a scale factor greater than 1.

There are two quadrilaterals of the same shape, with dashed rays connecting the corresponding vertices from the smaller shape to the larger shape.
Reduction

A decrease in size without changing the shape. This corresponds to a dilation with a scale factor between 0 and 1.

There are two quadrilaterals of the same shape, with dashed rays connecting the corresponding vertices from the larger shape to the smaller shape.

When performed on the coordinate plane, a dilation will have a specified scale factor as well as a specified center of dilation. If none is specified the origin is assumed to be the center of dilation.

Center of dilation

A fixed point on the coordinate plane about which a figure is either enlarged or reduced.

Two triangles which are dilations of each other have their corresponding vertices joined by dashed lines. The dashed lines are concurrent at a point, which is the center of dilation.

Coordinate form: The dilation \left(x,y\right) \to \left(kx,ky\right) takes the pre-image and dilates it by a factor of k, about the origin.

Function notation: The dilation D_{k,P}(A) takes the pre-image, A, and dilates it by a factor of k, with a center of dilation P.

If k>1, the dilation will be an enlargment, and if 0<k<1, the dilation will be a reduction. If k=1, the dilation maps the pre-image onto itself.

Worked examples

Example 1

Find the scale factor for the following dilation:

Two parallelograms are drawn. Parallelogram A B C D has side B C of length 16 and side D C of length 32. Segment A B and segment D C are marked parallel as well as segment B C and segment A D. Parallelogram A prime B prime C prime and D prime has side B prime C prime of length 2 and segment D prime C prime of length 4.

Approach

We can see that the pre-image has side lengths of 32 and 16, and the image has side lengths of 4 and 2. This indicates that the pre-image has been reduced. To find the scale factor we can divide one of the lengths of the image by the corresponding side length of the pre-image.

Solution

\dfrac{C'D'}{CD}=\dfrac{4}{32} Simplifying the quotient gives a scale factor of \dfrac{1}{8}.

Example 2

Dilate the figure using a scale factor of 4 with the origin as the center of dilation.

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Approach

Since the center of dilation is the origin, (0,0), we can take the coordinate of each point in the pre-image and multiply them by the scale factor to get the vertices of the image.

This means that:

(4, 8) \to (16, 32)

(10, 8) \to (40, 32)

(4, 14) \to (16, 56)

Solution

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Reflection

This transformation can be represented by the function notation D_{4,O} or by the coordinate form \left(x,y\right)\to\left(4x,4y\right).

Outcomes

MA.912.GR.2.1

Given a preimage and image, describe the transformation and represent the transformation algebraically using coordinates.

MA.912.GR.2.2

Identify transformations that do or do not preserve distance.

MA.912.GR.2.3

Identify a sequence of transformations that will map a given figure onto itself or onto another congruent or similar figure.

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