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Lesson Settings & Tools
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This lesson explores two new transformations, rotations and dilations. Rotations move shapes around a central point, while dilations either enlarge or shrink geometric figures without changing their shape.

### Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Explore

## Rotating Points About a Fixed Point

In the following applet, points and are movable. Arrange them as desired, then use the slider bar to rotate the points around the central point
Use the measuring tool to fill in the following table with the distances between and the image and preimage of each point. What conclusions can be drawn from the table? Repeat the process with different arrangements of points as needed to see a pattern.
Explore

## Rotating a Triangle About a Fixed Point

This time, use the slider bar to rotate triangle about point
Then, use the measuring tool to determine the angle measures of and Change the triangle and perform the same procedure. What can be said about rotations? Does the conclusion depend on the position of relative to the triangle?
Discussion

## Rotation

A rotation is a transformation in which a figure is turned about a fixed point The number of degrees the figure rotates is the angle of rotation. The fixed point is called the center of rotation. Rotations map every point in the plane to its image such that one of the following statements is satisfied.

• If is the center of rotation, then and are the same point.
• If is not the center of rotation, then and are equidistant from , with measuring
Rotations are usually performed counterclockwise unless stated otherwise.
Since rotations preserve side lengths and angle measures, they are rigid motions.
Discussion

## Rotation of a Point Around Another Point

Rotations can be performed by hand with the help of a straightedge, a compass, and a protractor.

To rotate point about point by an angle of measured counterclockwise, follow these five steps.

1
Draw
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Using the straightedge, draw the segment connecting the center of rotation and point
2
Place the Protractor
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Place the center of the protractor on and align it with

The protractor is placed as illustrated above when the rotation is counterclockwise. If the rotation has to be done clockwise, the protractor needs to be placed as follows.

3
Mark the Desired Angle
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Locate the corresponding measure on the protractor and make a small mark. In this case, the mark will be made at

4
Draw a Ray
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Using the straightedge, draw a ray with starting point that passes through the mark made in the previous step.
5
Draw Point
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Place the compass tip on and open it to the distance between and Without changing this setting and keeping the point of the compass at draw a small arc centered at that intersects the ray drawn before.
The intersection of the ray and the arc is the image after the give rotation.

Notice that this method of construction has also confirmed that is congruent to

Example

## Cyrptawheel

Vincenzo and Zosia are two friends who love exploring the world of puzzles. Within the pages of a book discussing transformations, they find a circular disk with letters on it. They decide to call the disk a Cyrptawheel. The Cryptawheel has a movable triangle with one vertex fixed at the center of the circular disk. To reveal its secrets, the friends need to rotate the triangle.
Two vertices of the triangle show the letters N and E. These are the first two letters of a clue. Rotate the triangle clockwise around its fixed vertex to find the other two letters. Zosia wants to perform this rotation using a protractor to be mathematically sure. Help her in this task.

### Hint

Start by rotating the vertices of the triangle. To do so, use a protractor to draw a angle.

### Solution

Zosia's goal is to find the image of the triangle after a clockwise rotation about its fixed vertex. Labeling the vertices of the triangle will make this rotation process simpler.

The center of rotation is and the angle of rotation is Perform the rotation by rotating one point at a time. To rotate place the center of the protractor on and align it with Use the protractor to draw a ray that starts from and makes a angle with

Then, mark a point on this ray so that is the same length as This is the image of after the rotation. Since is the radius of the inner circle, should also be on that circle.

Repeat the same process for the vertex to find Since is the center of rotation, will be in the same position as

Finally connect and to draw the image of after the rotation.

As shown, the image corresponds to P and corresponds to T. Therefore, the clue Vincenzo and Zosia is looking for is NEPT.

### Extra

Visualizing the Rotation
Vincenzo, on the other hand, completes this rotation task by solely relying on his eyes, without the use of any extra tools. The applet shows how he rotated the triangle about its fixed vertex.
Notice that a clockwise rotation and a counterclockwise rotation produce the same image. Therefore, there is no need to specify direction when rotating a figure by
Pop Quiz

## Practice Rotating Triangles

Rotate triangle around the red point with the specified angle and direction. Use the measuring tool to determine where the vertices of the triangle will end up after the rotation.

Discussion

## Rotating a Point Around the Origin at and

In the coordinate plane, when a point is rotated counterclockwise around the origin at certain angles, its coordinates change in a specific way. This occurs when the angle of rotation is or Try to figure these patterns out using the following applet.
From the diagram, the following rules can be set.
Counterclockwise Rotations Around the Origin
Angle of Rotation Rule
Example

## A Gateway to a New Realm

The clue Vincenzo and Zosia found reminded them of the word Neptune.

Vincenzo had previously heard stories about this magical place. He had even heard about three ordinary students who had found a door into a mystical library. Could this be another doorway to that place? As they got closer, a puzzle appeared between the columns of the gate.

External credits: macrovector
Graph the figure and its image after a counterclockwise rotation about the origin on the same coordinate plane.

### Hint

When a point with coordinates is rotated counterclockwise about the origin, its image becomes

### Solution

Zosia and Vincenzo want to rotate the figure counterclockwise around the origin. This can be achieved by rotating each point of the original figure. In this case, four points will be sufficient to determine the position of the image.

When a counterclockwise rotation is performed about the origin, the coordinates of the image can be written in relation to the coordinates of the preimage.

Rotation Rotation Rotation
Use the coordinate changes shown in the table that correspond to a counterclockwise rotation about the origin to determine the coordinates of the image of each point.
Now plot the image points and connect them. The final figure will be the image of the given figure after the rotation!

### Extra

Visualizing the Rotation
The applet shows how the figure is rotated counterclockwise about the origin. Notice that this rotation maps quadrilateral onto quadrilateral
Explore

## Exploring Dilations

The gate swings open as the figure rotates into place. Zosia takes the first step through the gate, while Vincenzo hesitantly waits to see what happens. Use the sliders to see how Zosia's size changes as she passes through the gate and down the hall.
External credits: Laitche, macrovector
As he watches Zosia's size change with amazement, Vincenzo notices that her image changes relative to a fixed point. How can this point be identified?
Illustration

## Dilating a Drawing

Feeling braver, Vincenzo steps through the gate as well. Inside, her and Zosia find an old notebook with a big red dot on one page. Zosia draws a figure on the page and watches as it changes size.
The friends are excited about the magical discovery and eager to comprehend the logic behind it. What other surprises lie ahead?
Discussion

## How to Dilate?

As they explore the magical realm, Vincenzo and Zosia come across Dilatius the Dimension Shifter, a wizard who can change the size of objects using dilations. They realize that the notebook they found belongs to the wizard and excitedly ask him to teach them about dilation. Dilatius thrilled to share his knowledge with them.

Concept

## Dilation

A dilation is a transformation that changes the size of a figure while keeping its shape the same. This transformation involves enlarging or reducing the figure by a certain length scale factor from a fixed point called the center of dilation. For example, the image of every point on a leaf lies on the ray that starts at the center of the dilation and passes through its preimage.
As shown in the diagram, is the center of the dilation, is the scale factor, is the preimage, and is the image point. By definition, the scale factor can also be defined as the ratio of a length in the image to the corresponding length in the preimage.

When the scale factor is greater than the dilation is called an enlargement because the image is larger than the preimage. When the scale factor is between and the dilation is called a reduction because the image is smaller than the preimage.
Discussion

## Coordinate Rule for Dilations

When a point is dilated using a scale factor of and a center of dilation at the origin, the coordinates of its image are found by multiplying the coordinates of the preimage by

The diagram shows how the image changes as the preimage and the scale factor change.

Example

## Practicing Dilatius's Teachings

Dilatius is impressed by Vincenzo and Zosia's eagerness to learn. They seem to have picked up the dilation spell using the coordinate rule quickly, so he challenges them to dilate the following triangle.

Draw the image of the triangle after a dilation with center and a scale factor of

### Hint

To find the image of a vertex after a dilation with scale factor multiply its coordinates by

### Solution

Zosia and Vincenzo need to dilate the triangle using a scale factor of with respect to the origin. Start by identifying the vertices of the triangle.

When the center of dilation is the origin, each coordinate of the preimage is multiplied by the scale factor to find the coordinates of the image.
Find the coordinates of the vertices of after a dilation with a scale factor
Dilation With Scale Factor
Preimage Multiply by Image
Finally, plot the image points and connect them with segments. The new triangle will be the image of the given figure after the dilation!

Visualizing the Dilation
To check the answer, draw rays from the origin through the vertices of the original figure. The vertices of the dilation should lie on those rays.
Notice that is times as long as since is the scale factor. This also applies to other side lengths.
Example

## Dilatius's Final Lesson

Dilatius teaches Vincenzo and Zosia to use reducio to make things smaller and enlargio to make them bigger. These phrases produce a reduction and an enlargement, respectively. He then quizzes them about the magic behind the drawings in his notebook.

The green square is a dilation of the blue square. Determine whether the dilation is an enlargement or a reduction and the scale factor of the dilation.

### Hint

The scale factor is the ratio of the sides lengths of the image to the corresponding side lengths of the original figure.

### Solution

There are two types of dilations.

1. Enlargement: The image is larger than the original figure. An enlargement is the product of a scale factor greater than
2. Reduction: The image is smaller than the original figure. A reduction is the product of a scale factor between and

In the given coordinate plane, it can be seen that the green square is smaller than the blue square. This means that the dilation is a reduction.

Next, remember that the scale factor is the ratio of the sides lengths of the image to the corresponding side lengths of the preimage.
In the graph, is units long and the corresponding side is units long.
Substitute these values into the formula to find the scale factor.
The scale factor is equal to Therefore, the green square is a reduction of the blue square with a scale factor of The answer is C.