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# Describing Dilations

Concept

## Dilation

A dilation is a transformation that enlarges or reduces the size of a figure. Therefore, the image created will have a different size than the preimage. However, a dilation preserves angle measures and the ratio between corresponding parts.

Concept

### Center of dilation

The center of dilation is a point that regulates where the image of the figure is placed.

From the center of dilation, $C,$ segments are drawn through the vertices of the preimage.

The lengths between $C$ and the vertices of the figure will then be scaled by the scale factor. This will determine if the image is placed closer or further from $C$ than the preimage. The center of dilation could also be a vertex of the figure. In that case, some of the segments drawn will only be extensions of the segments in the figure.

Concept

### Scale factor

In a dilation, the scale factor refers to the ratio between the corresponding sides of the preimage and the image. Therefore, the scale factor, $k,$ can be calculated by dividing the length of one part in the image with the corresponding part in the preimage. The length of the preimage part should always be in the denominator as it belongs to the original figure.
When dilating a figure, the different parts of the figure are multiplied by the scale factor.
Concept

### Enlargement and reduction

Depending on the scale factor, the image will become bigger or smaller than the preimage. If the scale factor is greater than $1,$ the sides of the image will be longer than the sides of the preimage. This is called an enlargement, and it will place the image further away from the center of dilation than the preimage.
Reset

As opposed to an enlargement, if the scale factor is a positive number less than $1$ the image will have shorter sides that the preimage. This is a reduction and moves the image closer to $C$ than the preimage.
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Exercise

The MBA, Mathleaks' Basketball Association, has become very popular among kids. Therefore, a smaller version of the official basketball has been created.

Determine the preimage and image, the scale factor and if the dilation is an enlargement or a reduction.

Show Solution
Solution

The preimage can be determined by identifying which of the balls is the original figure and which is the copy. Since a smaller version of the official ball is being created, we know the larger one is the preimage.

The scale factor can be determined by finding the ratio between two corresponding parts of the preimage and the image. In this case, the radii for both basketballs are known.

Now, the scale factor, $k,$ can be calculated by dividing the radius of the image with that of the preimage.
$k=\dfrac{\text{image radius}}{\text{preimage radius}}$
$k=\dfrac{{\color{#0000FF}{3.6}}}{{\color{#009600}{4.8}}}$
$k=0.75$
The scale factor is $k=0.75.$ Since $k<1,$ the dilation is a reduction. This make sense because we know they want to create a smaller basketball.
Method

## Dilating a Figure in a Coordinate Plane

When dilating a figure in the coordinate plane, the coordinates of the figure's vertices are multiplied by the scale factor. The new coordinates, the image's vertices, can then be graphed in the coordinate plane.

The quadrilateral $ABCD$ has the vertices $A(1,2),$ $B(3,1),$ $C(3,\text{-}1)$ and $D(1,\text{-}1)$ and is dilated by a scale factor of $3.$ Note that when dilating a figure in a coordinate plane, the center of dilation is the origin unless otherwise stated.

### 1

Multiply the coordinates with the scale factor

By multiplying the coordinates of the vertices $A,$ $B,$ $C$ and $D$ with the scale factor, $3,$ the positions for the image's vertices can be found. Each $x$- and $y$-coordinate is individually multiplied by $3$.

Preimage $\left(3x,3y \right)$ Image
$A(1,2)$ $(3\cdot1,3\cdot2)$ $A'(3,6)$
$B(3,1)$ $(3\cdot3,3\cdot1)$ $B'(9,3)$
$C(3,\text{-}1)$   $(3\cdot3,3(\text{-}1))$ $C'(9,\text{-}3)$
$D(1,\text{-}1)$   $(3\cdot1,3(\text{-}1))$ $D'(3,\text{-}3)$

The images $A',$ $B',$ $C'$ and $D'$ are the vertices of the image.

### 2

Plot the vertices of the image in the coordinate plane

The images vertices $A'(3,6),$ $B'(9,2),$ $C'(9,\text{-}3)$ and $D'(3,\text{-}3)$ can be plotted in the coordinate plane.

### 3

Draw the image by connecting the vertices

Finally, the image can be drawn by connecting the image's vertices with segments.

The quadrilateral $A'B'C'D'$ has the same shape as the preimage $ABCD$ but the corresponding sides are three times longer. This makes the dilation an enlargement and since the center of dilation is the origin, the image is further from the origin than the preimage.

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Exercise

Dilate the line segment with a scale factor of $\frac{1}{4}.$ Is the image longer or shorter than the preimage?

Show Solution
Solution

To dilate the segment $\overline{AB},$ the coordinates of $A$ and $B$ can be multiplied with the scale factor $\frac{1}{4}.$ The coordinates can be read from the coordinate plane.

The coordinates are $A(\text{-}2,\text{-}2)$ and $B(1,4).$ Now, take the $x$- and $y$-values and multiply them by $\frac{1}{4}.$ Note that multiplying a number by $\frac{1}{4}$ is the same as multiplying by $0.25.$

Preimage $(0.25x, 0.25y)$ Image
$A(\text{-}2,\text{-}2)$ $({\color{#0000FF}{0.25}} (\text{-}2),{\color{#0000FF}{0.25}}(\text{-} 2))$ $A'(\text{-}0.5,\text{-}0.5)$
$B(1,4)$ $({\color{#0000FF}{0.25}} \cdot 1,{\color{#0000FF}{0.25}} \cdot 4)$ $B'(0.25,1)$

We have now found the coordinates of the image's endpoints: $A'(\text{-}0.5,\text{-}0.5)$ and $B'(0.25,1).$ We'll mark them in the coordinate plane and connect them to draw the image.

Notice that the image, $\overline{A'B'},$ is shorter than the preimage, $\overline{AB}.$ This is because $\frac{1}{4}<1.$ Therefore, when the preimage was dilated, it was moved closer to the center of dilation. A dilation of a line segment will always create parallel lines if the center of dilation isn't on the line.

## Dilating a Figure

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A figure can be dilated using a compass and a straightedge.

The figure $ABCD$ should be dilated with the point $P$ as the center of dilation and a scale factor of $4.$ Using a straightedge, rays are drawn from $P$ through the vertices of the quadrilateral.

A dilation scales the lengths between the center of dilation and the vertices of the figure with a scale factor, creating an image of the figure. Using the scale factor $4,$ the dilation will be an enlargement. To be able to scale the segments, the distance of each vertex from the center is measured with a compass. Place the needle point at $P$ and the pencil at one of the figure's vertices.

With the scale factor $4,$ the length from $P$ to the image vertex $A'$ should be $4$ times the length of $\overline{PA}.$ By moving the compass three times along the ray $\overrightarrow{PA},$ keeping the same length on the compass, the position for $A'$ is found. Let the pencil of the last compass mark the position.

This process should now be repeated for the vertices $B,$ $C$ and $D.$ When all of the image's vertices are marked, it's possible to draw the image.

Finally, segments are drawn between the image vertices $A'$, $B',$ $C'$ and $D',$ with a straightedge, creating the quadrilateral $A'B'C'D'.$

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