Topics
6. Polygons
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United States of AmericaVirginia
Grade 7

6.04 Dilations in the coordinate plane

Lesson
Practice

Dilations in the coordinate plane

We've learned that similar polygons 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} has side lengths of 3\text{ cm}, 4\text{ cm}, and 5\text{ cm}. If it is dilated by a scale factor of 2 to produce \triangle {XYZ}, then \triangle {XYZ} will have side lengths of 6\text{ cm}, 8\text{ cm}, and 10\text{ cm}, as shown:

A triangle with side lengths of 3, 4, and 5 units, and a larger triangle with side lengths of 6, 8, and 10 units.

\frac{6}{3}=\frac{8}{4}=\frac{10}{5}={2}

If any of these ratios were not equal to 2, then this would not be a dilation.

Well, we need two things:

  1. A center of dilation: a point from where we start the enlargement. This may be inside or outside the original shape, and for now we will only use the origin on a coordinate plane.

  2. A scale factor: the ratio by which we increase or decrease the shape. We calculate a scale factor just like we would calculate the ratio of the sides in similar triangles.

Exploration

The applet allows us to see the image for a rectangle we choose along with the scale factor of our choice.

Use the slider to select the desired scale factor and the three blue points on the preimage rectangle to make your preimage.

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  1. Can you make a preimage where the point \left(0,\,0\right) is inside the image? outside the image? on an edge of the image? How?

  2. Can you make a preimage which is completely inside the image? How?

  3. What scale factors make the image larger than the preimage? Which make is smaller? Which make it the same size?

A scale factor can increase or decrease the size of the new shape, called the image. The original shape before the dilation is called the preimage.

For example, scale factor of 3 means the image will have side lengths 3 times as big, whereas a scale factor of \dfrac{1}{2} means the image will have side lengths \dfrac{1}{2} as big as the original.

In general,

  • If the scale factor, k, has k \gt 1, the image will be larger than the preimage

  • If the scale factor, k, has 0 \lt k \lt 1, the image will be smaller than the preimage

We can use the coordinates of the vertices of polygons on the coordinate plane to find the image. We label points on the preimage with letters like A,\,B,\, and C. We label points on the image with prime notation like A',\,B',\, and C' (which we read as "A prime, B prime, and C prime) using the same corresponding letters from the preimage.

Consider the following image which dilates the green preimage by a scale factor of 3 from the origin. Let's have a look at the coordinates of the vertices of the rectangles.

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The dilation is by a scale factor of 3, and the center of dilation is at \left(0,\,0\right).

PreimageImage
A(1,2)A'(3,6)
B(1,0)B'(3,0)
C(2,0)C'(6,0)
D(2,2)D'(6,2)

What do we notice about both the x and the y coordinates of the preimage and image?

With a scale factor of k and a center of dilation \left(0,\,0\right), the preimage point \left(x,\,y\right) will become the image point of \left(kx,\,ky\right).

The ratios of corresponding side lengths must be the same, for example in a rectangle:

\frac{A'B'}{AB}=\frac{B'C'}{BC}=\frac{C'D'}{CD}=\frac{D'A'}{DA}

Examples

Example 1

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

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Worked Solution
Create a strategy

Compare the ratios of corresponding side lengths.

Apply the idea

Sides A'B' and AB, and B'C' and BC are corresponding sides of the rectangles.

Length of A'B':

\displaystyle A'B'\displaystyle =\displaystyle {15-6}Subtract the x-coordinate of A' from the x-coordinate of B'
\displaystyle =\displaystyle 9Evaluate

Length of AB:

\displaystyle AB\displaystyle =\displaystyle 8-3Subtract the x-coordinate of A from the x-coordinate of B
\displaystyle =\displaystyle 5Evaluate

Length of B'C':

\displaystyle B'C'\displaystyle =\displaystyle {18-8}Subtract the y-coordinate of B' from the y-coordinate of C'
\displaystyle =\displaystyle 10Evaluate

Length of BC:

\displaystyle BC\displaystyle =\displaystyle 9-4Subtract the y-coordinate of B from the y-coordinate of C
\displaystyle =\displaystyle 5Evaluate
\displaystyle \frac{A'B'}{AB}\displaystyle =\displaystyle \frac{9}{5}Substitute A'B'=9, AB=5
\displaystyle \frac{B'C'}{BC}\displaystyle =\displaystyle \frac{10}{5}Substitute B'C'=10, and BC=5
\displaystyle \frac{A'B'}{AB}\displaystyle \neq\displaystyle \frac{B'C'}{BC}Compare the ratios

Rectangle A'B'C'D' is not a dilation of rectangle ABCD.

Example 2

Dilate the figure by the given factor using the origin as the center of dilation:

a

Identify the coordinates of the image and graph the image with a scale factor of 3.

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Worked Solution
Create a strategy

Since the origin is the center of dilation, we can multiply each x and y-coordinate by the scale factor to dilate the preimage.

Apply the idea

Point A(1,\,1) dilated by a scale factor of 3 becomes A'(3,\,3).

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Point B(3,\,1) dilated by a scale factor of 3 becomes B'(9,\,3).

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Point C(3,\,3) dilated by a scale factor of 3 becomes C'(9,\,9).

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Point D(1,\,3) dilated by a scale factor of 3 becomes D'(3,\,9).

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The coordinates of the image A'B'C'D' are A'(3,\,3),\,B'(9,\,3),\,C'(9,\,9),\, and D'(3,\,9).

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Reflect and check

Compare the ratios of the corresponding side lengths to check and confirm the scale factor of 3.

Sides A'B' and AB, and B'C' and BC are corresponding sides of the rectangles.

Length of A'B':

\displaystyle A'B'\displaystyle =\displaystyle {9-3}Subtract the x-coordinate of A' from the x-coordinate of B'
\displaystyle =\displaystyle 6Evaluate

Length of AB:

\displaystyle AB\displaystyle =\displaystyle 3-1Subtract the x-coordinate of A from the x-coordinate of B
\displaystyle =\displaystyle 2Evaluate

Length of B'C':

\displaystyle B'C'\displaystyle =\displaystyle {9-3}Subtract the y-coordinate of B' from the y-coordinate of C'
\displaystyle =\displaystyle 6Evaluate

Length of BC:

\displaystyle BC\displaystyle =\displaystyle 3-1Subtract the y-coordinate of B from the y-coordinate of C
\displaystyle =\displaystyle 2Evaluate
\displaystyle \frac{A'B'}{AB}\displaystyle =\displaystyle \frac{6}{2}Substitute A'B'=6, AB=2
\displaystyle \frac{B'C'}{BC}\displaystyle =\displaystyle \frac{6}{2}Substitute B'C'=6, and BC=2
\displaystyle \frac{A'B'}{AB}\displaystyle =\displaystyle \frac{B'C'}{BC}Compare the ratios

Rectangle A'B'C'D' is a dilation of rectangle ABCD by a scale factor of 3.

b

Identify the coordinates of the image and graph the image with a scale factor of \dfrac{1}{2}.

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Worked Solution
Create a strategy

Since the origin is the center of dilation, we can multiply each x and y-coordinate by the scale factor to dilate the preimage.

Apply the idea

Point P(2,\,4) dilated by a scale factor of \dfrac{1}{2} becomes P'(1,\,2).

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Point Q(4,\,8) dilated by a scale factor of \dfrac{1}{2} becomes Q'(2,\,4).

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Point R(6,\,4) dilated by a scale factor of \dfrac{1}{2} becomes R'(3,\,2).

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Point S(4,\,2) dilated by a scale factor of \dfrac{1}{2} becomes S'(2,\,1).

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Point T(2,\,2) dilated by a scale factor of \dfrac{1}{2} becomes T'(1,\,1).

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The coordinates of the image P'Q'R'S'T' are P'(1,\,2),\,Q'(2,\,4),\,R'(3,\,2),\,S'(2,\,1),\, and T'(1,\,1).

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Reflect and check

Compare the ratios of the corresponding side lengths to check and confirm the scale factor of \dfrac{1}{2}.

Sides P'T' and PT, and T'S' and TS are corresponding sides of the rectangles.

Length of P'T':

\displaystyle P'T'\displaystyle =\displaystyle {2-1}Subtract the y-coordinate of P' from the y-coordinate of T'
\displaystyle =\displaystyle 1Evaluate

Length of PT:

\displaystyle PT\displaystyle =\displaystyle 4-2Subtract the y-coordinate of P from the y-coordinate of T
\displaystyle =\displaystyle 2Evaluate

Length of T'S':

\displaystyle T'S'\displaystyle =\displaystyle {2-1}Subtract the x-coordinate of T' from the x-coordinate of S'
\displaystyle =\displaystyle 1Evaluate

Length of TS:

\displaystyle TS\displaystyle =\displaystyle 4-2Subtract the x-coordinate of T from the x-coordinate of S
\displaystyle =\displaystyle 2Evaluate
\displaystyle \frac{P'T'}{PT}\displaystyle =\displaystyle \frac{1}{2}Substitute P'T'=1, PT=2
\displaystyle \frac{T'S'}{TS}\displaystyle =\displaystyle \frac{1}{2}Substitute T'S'=1, and TS=2
\displaystyle \frac{P'T'}{PT}\displaystyle =\displaystyle \frac{T'S'}{TS}Compare the ratios

Polygon P'Q'R'S'T' is a dilation of polygon PQRST by a scale factor of \dfrac{1}{2}.

Example 3

Emma and Noah are designing a garden layout. They measured their garden and are trying to make a scale model to plan the placement of plants and pathways. Their garden includes a section for a rectangular vegetable garden and a square flower bed on the corner to attract pollinators.

The image shows 2 composite shapes labeled as 'Real garden' and 'Scale drawing'. Ask your teacher for more information.
a

What is the scale factor from the garden to the drawing?

Worked Solution
Create a strategy

Convert both dimensions to the same units and compare the ratio of the corresponding side lengths to determine the scale factor.

Apply the idea

The actual length of the garden is 6\text{ m} and the length in the drawing is 15\text{ cm}. We can use these two side lengths to create a ratio.

To convert the length of 6\text{ m} to \text{cm} we need to multiply by 100.6\text{ m} \cdot 100 = 600 \text{ cm}

\displaystyle \frac{\text{Drawing Length}}{\text{Actual Length}}\displaystyle =\displaystyle \frac{15 \text{ cm}}{600 \text{ cm}}Divide corresponding lengths
\displaystyle =\displaystyle \dfrac{1}{40}Simplify the fraction

The scale factor from the actual garden to the scaled down drawing is \dfrac{1}{40}.

b

Draw the scale model with all sides labeled.

Worked Solution
Create a strategy

Scale down all dimensions by the scale factor of \dfrac{1}{40} after converting to the same units.

Apply the idea
The image shows a composite shape with measurements of 10cm, 15cm, and 2.5cm. Ask your teacher for more information.

First, to convert from meters to centimeters we need to multiply each actual length by 100.

4\text{ m} \cdot 100 = 400\text{ cm}

1\text{ m} \cdot 100 = 100\text{ cm}

Then, we need to use the scale factor to dilate these actual lenghts down to the drawing lengths.

400\text{ cm} \cdot \dfrac{1}{40} = 10 \text{ cm}

100\text{ cm} \cdot \dfrac{1}{40} = 2.5 \text{ cm}

The image shows a composite shape with measurements of 10cm, 15cm, 15cm, 2.5cm, 2.5cm, and 2.5cm. Ask your teacher for more information.

We know the opposite length of the rectangular vegetable garden because it will be equivalent to the opposite side. Same goes for the square flower bed, because we know the shape is a square all the sides are equal.

The image shows a composite shape with measurements of 10cm, 15cm, 17.5cm, 2.5cm, and 2.5cm. Ask your teacher for more information.

Instead of having those individual lengths listed on the side where the vegetable garden and flower bed are adjacent, we can total those lengths.

The image shows a composite shape with measurements of 10cm, 15cm, 17.5cm, 2.5cm, 2.5cm, and 7.5cm. Ask your teacher for more information.

We can find the final missing length by subtracting the side length of the flower bed from the width of the vegetable garden.

10\text{cm} - 2.5\text{cm} = 7.5\text{cm}

Idea summary

A scale factor can increase or decrease the size of the new shape.

  • If the scale factor, k, has k \gt 1, the image will be larger than the preimage

  • If the scale factor, k, has 0 \lt k \lt 1, the image will be smaller than the preimage

With a scale factor of k and a center of dilation \left(0,\,0\right), the preimage point \left(x,\,y\right) will become the image point of \left(kx,\,ky\right).

The ratios of corresponding side lengths must be the same, for example in a rectangle:

\frac{A'B'}{AB}=\frac{B'C'}{BC}=\frac{C'D'}{CD}=\frac{D'A'}{DA}

Outcomes

7.MG.4

The student will apply dilations of polygons in the coordinate plane.

7.MG.4a

Given a preimage in the coordinate plane, identify the coordinates of the image of a polygon that has been dilated. Scale factors are limited to 1/4,1/2, 2, 3, or 4. The center of the dilation will be the origin.

7.MG.4b

Sketch the image of a dilation of a polygon limited to a scale factor of 1/4,1/2, 2, 3, or 4. The center of the dilation will be the origin.

7.MG.4c

Identify and describe dilations in context including, but not limited to, scale drawings and graphic design.

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