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7.01 Dilations

Lesson

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).

Example 3

Identify whether the following transformation preserves: distance, angles, or distance and angles.

A translation to the right 3 units, followed by a dilation using a scale factor of 2.

Approach

One way to approach this problem is by sketching an example so we can see how an object will change based on the transformation done to it. Let's use a 2 \times 2 square on the coordinate plane as an example.

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Solution

2
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After we translate the square 3 units to the right we get this image.

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After the dilation by a scale factor of 2 with center of dilation of the bottom left vertex, we get this image.

We can see that distance has not been preserved since the side lengths of the image are larger than the preimage, however the angle measurements appear to not have changed. Therefore a translation to the right 3 units, followed by a dilation using a scale factor of 2 preserves only angle measurements.

Reflection

We can be more confident that the angle measures have not changed if we do a construction using technology.

Geometry calculator tool with square ABCD plotted with point A at 3,0 point B at 5,0 point C at 5,2 and point D at 3,2

First we can construct the image after the translation, but before the dilation using the Polygon tool.

Geometry calculator tool with square ABCD plotted with each angle measure labeled as 90 degrees

Then we can measure all of the angles in this polygon by selecting the Angle tool and clicking somewhere in the middle of the polygon.

Geometry calculator tool with square ABCD plotted with a larger square A prime B prime C prime D prime plotted. The two squares share a vertex A and A prime at 3,0. B prime is at 7,0. C prime is at 7,4. And D prime is at 3,4.

Next, we can using the Dilate from Point tool and click on the polygon, then the point A and then type in 2 to perform the required dilation.

Geometry calculator tool with squares ABCD and A prime B prime C prime D prime plotted. All 4 angles of both squares are labeled 90 degrees

Finally, we can use the Angle tool to find all of the angle measures on the final image.

We can then explore this construction by moving around the points on ABCD to see what happens to A'B'C'D'.

Outcomes

G.SRT.A.1

Use properties of dilations given by a center and a scale factor to solve problems and to justify relationships in geometric figures.

G.MP1

Make sense of problems and persevere in solving them.

G.MP3

Construct viable arguments and critique the reasoning of others.

G.MP4

Model with mathematics.

G.MP5

Use appropriate tools strategically.

G.MP6

Attend to precision.

G.MP7

Look for and make use of structure.

G.MP8

Look for and express regularity in repeated reasoning.

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