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When learning about transformations, rotations were mentioned and used briefly. In this lesson, rotations will be studied more deeply by analyzing the relationships between a point

Q

and its image

Q^{'}

under a rotation.

Catch-Up and Review

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

Transformations
Rigid motion
Composition of transformations
Angle
Coordinate plane
Perpendicular bisector

Explore

Rotating Points About a Center

In the following applet, points

A,

B,

C,

and

D

can be moved and rotated about point

P .

Rotating Points About a Point

Make an arrangement of points and rotate them about

P .

Then, find the distance between

P

and each preimage, and the distance between

P

and each image.

P A P C = ? = ? P A^{'} P C^{'} = ? = ? P B P D = ? = ? P B^{'} P D^{'} = ? = ?

Next, make a different arrangement, rotate the points, and find the distances. What can be said about rotations? If needed, make more arrangements and rotations.

Explore

Rotating a Triangle About a Center

Perform a rotation to

△ J K L

about point

P .

Use a protractor to find the measure of

∠ J P J^{'},

∠ K P K^{'},

and

∠ L P L^{'} .

Rotating a Triangle About a Point

Change the triangle and perform the same process. What can be said about rotations? Does the conclusion depend on the position of

P

relative to the triangle?

Discussion

Rotations Along Circles

The first exploration shows the preimage and the image of a point under a rotation are the same distance from the center of rotation. That is, the image moves along a circle passing through the preimage and centered at the center of rotation.

Rotating Points About a Point

The second exploration shows that after performing a rotation, the angles formed by each preimage, the center of rotation, and the corresponding image all have the same measure. With these properties in mind, rotations can be properly defined.

Discussion

Defining a Rotation

Identifying whether one figure is the image of another figure under rotation can be difficult. A key aspect to observe is whether the center of rotation is the same distance from an image as it is from its preimage.

Concept

Rotation

A rotation is a transformation in which a figure is turned about a fixed point $P$ by a certain angle measure $α^{\circ} .$ The point $P$ is called the center of rotation. Rotations map every point $A$ in the plane to its image $A^{'}$ such that one of the following statements is satisfied.

If $A$ is the center of rotation, then $A$ and $A^{'}$ are the same point.
If $A$ is not the center of rotation, then $A$ and $A^{'}$ are the same distance from $P,$ and $∠ A P A^{'}$ has a measure of $α^{\circ} .$

The angle formed by a preimage, the center of rotation, and the image is called the angle of rotation and its measure is

α^{\circ} .

Since rotations preserve side lengths and angle measures, they are rigid motions.

Rotation of point A around center P

Usually, rotations are performed counterclockwise unless otherwise stated.

Example

Identifying Rotations

Consider the following three triangles and a point

P .

Use the given measuring tool to find the distance from each vertex to

P

and the angles formed by each preimage, the point

P,

and the corresponding image.

Image of a Triangle After a Rotation

Which of the triangles,

△ A^{'} B^{'} C^{'}

or

△ A^{''} B^{''} C^{''}

, is the image of

△ A B C

under a rotation about

P ?

If one of the triangles is a rotation of

△ A B C

about

P,

what is the measure of the angle of rotation?

Hint

Remember, after performing a rotation, the preimage and the image of a point are the same distance from the center of rotation. The angle of rotation is formed by a preimage, the center of rotation, and the corresponding image.

Solution

Remember that, after performing a rotation, the preimage and the image of a point are the same distance from the center of rotation. Then, start by finding the distances between each vertex and

P .

Notice that the vertices of

△ A^{'} B^{'} C^{'}

are further from

P

than the vertices of

△ A B C .

Consequently,

△ A^{'} B^{'} C^{'}

cannot be the image of

△ A B C

under a rotation about

P .

Next, find and compare the measures of

∠ A P A^{'},

∠ B P B^{'},

and

∠ C P C^{'} .

As can be seen,

∠ A P A^{'},

∠ B P B^{'},

and

∠ C P C^{'}

have all the same measure, which is

15 0^{\circ}

when measured counterclockwise or

21 0^{\circ}

when measured clockwise. Therefore,

△ A^{''} B^{''} C^{''}

is the image of

△ A B C

under a rotation about

P .

The angle of rotation is either

15 0^{\circ}

or

21 0^{\circ} .

Discussion

Rotations Performed by Hand

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

Points P and A

To rotate point $A$ about point $P$ by an angle of $13 0^{\circ}$ measured counterclockwise follow these five steps.

1

Draw

\overline{P A}

Using a straightedge, draw the segment connecting the center of rotation and point

A .

Drawing the segment connecting P and A

2

Place the Protractor

Place the center of the protractor on $P$ and align it with $\overline{P A} .$

Placing the protractor on segment PA so that the rotation is counterclockwise

The way the protractor was placed before works when the rotation is counterclockwise. If the rotation has to be done clockwise, the protractor needs to be placed as follows.

Placing the protractor so that the rotation is clockwise

3

Mark the Desired Angle

Locate the corresponding measure on the protractor and make a small mark. In this case, the mark will be made at $13 0^{\circ} .$

Making a mark at 130 degrees

4

Draw a Ray

Using the straightedge, draw a ray with starting point

P

that passes through the mark made in the previous step.

Drawing a Ray starting a P that makes a 130 degree angle with segment PA

5

Draw Point

A^{'}

Place the compass tip on

P

and open it to the distance between

P

and

A .

With this setting, make an arc centered at

P

that intersects the ray drawn before.

Measuring PA with the compass and making an arc that intersects the ray drawn before

The intersection between the ray and the arc is the image of

A

under the give rotation. Notice that, by construction,

P A

is equal to

P A^{'} .

Points P, A, and A' (image of A under a 130 degree rotation about P)

Example

Rotating a Triangle

On a geometry test, Ignacio was asked to perform a $7 0^{\circ}$ counterclockwise rotation to $△ A B C$ about point $P .$

Rotating a Triangle About a Point

Draw $△ A B C$ and its image under this rotation.

Answer

Hint

Rotate the vertices of $△ A B C$ one at a time. Then, draw the triangle formed by the three images.

Solution

To rotate

△ A B C

, the rotation can be performed on each vertex, one at a time. For example, start by rotating

A .

To do so, first draw

\overline{P A}

using a straightedge.

Then, place the center of the protractor on

P

and align it with

\overline{P A}

in such a way that the rotation is counterclockwise. Then, make a small mark at

7 0^{\circ} .

Next, draw a ray with starting point

P

that passes through the mark made before.

Finally, place the compass tip on

P

and open the compass to the distance between

P

and

A .

With this setting, make an arc that intersects the ray. The intersection point is the image of

A

under the rotation.

Vertices

B

and

C

can be rotated following the same steps.

Finally, the image of $△ A B C$ under the given rotation is the triangle formed by $A^{'},$ $B^{'},$ and $C^{'} .$

Pop Quiz

Practice Rotations

Rotate $△ A B C$ about point $P$ by the specified angle measure. To do so, place points $A^{'},$ $B^{'},$ and $C^{'}$ where they should be after the rotation. Use the measuring tool if needed.

Performing random rotations to random triangles

Discussion

Rotating Points by $9 0^{\circ},$ $18 0^{\circ},$ and $27 0^{\circ}$

In the coordinate plane, there is a particular relationship between the coordinates of a point and those of its image after a counterclockwise rotation about the origin. This relation occurs when the angle of rotation is either

9 0^{\circ},

18 0^{\circ},

or

27 0^{\circ} .

Try to figure it out by using the following applet.

Rotations of a points about the origin

From the diagram, the following relations can be set.

The image of $(a, b)$ under a $9 0^{\circ}$ rotation about the origin is $(- b, a) .$
The image of $(a, b)$ under a $18 0^{\circ}$ rotation about the origin is $(- a, - b) .$
The image of $(a, b)$ under a $27 0^{\circ}$ rotation about the origin is $(b, - a) .$

Example

Finding the Center of Rotation

Given a figure and its image under a rotation, the following theorem can be used to find the center of rotation.

Converse Perpendicular Bisector Theorem
If a point is equidistant from the endpoints of a line segment, then it lies on the perpendicular bisector of the segment.

With this theorem in mind, consider the following example. In the diagram, quadrilateral $A^{'} B^{'} C^{'} D^{'}$ is the image of $A B C D$ under a certain rotation.

Preimage of a Quadrilateral After a Rotation

Find the center and angle of rotation.

Answer

Angle of Rotation: $12 0^{\circ}$ clockwise or $24 0^{\circ}$ counterclockwise.
Graph:

Hint

Remember that the center of rotation is equidistant from the preimage and the image of each vertex. Use the Converse Perpendicular Bisector Theorem. The center is the intersection point between two perpendicular bisectors.

Solution

The first step is to find the center of rotation. Remember, by definition, a point and its image under a rotation are the same distance from the center.

The center of rotation is equidistant from a point and its image.

Therefore, by the Converse of the Perpendicular Bisector Theorem, the center lies on the perpendicular bisector of $\overline{A A^{'}},$ for instance. Then, with the aid of a compass and a straightedge, start by constructing the perpendicular bisector of this segment.

To determine the center's exact position, draw a second segment joining a vertex and its image, for example, $\overline{D D^{'}} .$ Then, draw the perpendicular bisector of this segment. The intersection between both perpendicular bisectors is the center of rotation.

Notice that drawing only two perpendicular bisectors is enough to find the center of rotation because all will intersect at the same point. Since the sense of rotation was not specified, both measures will be found using a protractor.

The angle of rotation is either $24 0^{\circ}$ counterclockwise or $12 0^{\circ}$ clockwise.

Pop Quiz

Finding the Center and Angle of a Rotation

In the following applet, the solid triangle is the image of the dashed triangle under a certain counterclockwise rotation. Place the point where the center of the rotation should be. Then, select the appropriate angle of rotation.

Performing random rotations to random triangles

Extra

About the Applet

If the point is not placed close enough to the center of rotation, when the Check Answer button is pushed, a red area is highlighted indicating the region where the center of rotation is located.

Example

Performing a Composition of Rotations

Recall that rotations are transformations and that transformations can be composed. Therefore, it is possible to have a composition of two or more rotations. On a geometry exercise, the following two rotations are given.

$R_{1} :$ $9 0^{\circ}$ counterclockwise rotation about $C_{1} (0, 0) .$
$R_{2} :$ $18 0^{\circ}$ counterclockwise rotation about $C_{2} (0, - 1) .$

LaShay has to perform both rotations to $△ A B C,$ one after the other, but the book does not indicate the composition's order.

Triangle with vertices A(-4,1), B(-1,2), and C(-3,4)

According to the book, the correct coordinates of

B^{''}

are

(2, - 1) .

Knowing this, help LaShay to find the coordinates of

C^{''}

and draw

△ A^{''} B^{''} C^{''} .

Answer

Hint

Recall that a $9 0^{\circ}$ counterclockwise rotation about the origin maps point $(a, b)$ onto $(- b, a) .$

Solution

Start by applying any of the given two rotations. For example, apply

R_{1}

first. Notice it is a

9 0^{\circ}

counterclockwise rotation about the origin. Then, it maps

(a, b)

onto

(- b, a) .

Knowing this, the image of

A,

B,

and

C

can be quickly found.

A (- 4, 1) B (- 1, 2) C (- 3, 4) \to \to \to A^{'} (- 1, - 4) B^{'} (- 2, - 1) C^{'} (- 4, - 3)

Therefore, the image of

△ A B C

under

R_{1}

looks as follows.

Next, apply

R_{2}

to

△ A^{'} B^{'} C^{'} .

Remember, it is a

18 0^{\circ}

counterclockwise rotation about

C_{2} (0, - 1) .

Since the coordinates of

B^{''}

are the same as the book says, the rotations were applied in the correct order. Consequently, the coordinates of

C^{''}

are

(4, 1) .

Notice that applying the rotations in the reverse order would produce a different image.

Discussion

Composition of Rotations About Different Centers

Think about the preimage and the final image obtained by LaShay in the previous example. Is it possible to map

△ A B C

onto

△ A^{''} B^{''} C^{''}

performing only one rotation? The answer is yes! A

27 0^{\circ}

counterclockwise rotation about

(- 1, - 1)

does it.

Finding center and angle of rotation that maps triangle ABC onto triangle A''B''C''

In general, the composition of two rotations is a rotation, except if the sum of their angles of rotation is a multiple of

36 0^{\circ} .

Investigate what happens in this case by using the following applet.

Place the centers $A$ and $B$ at any place within the square formed by the axes and the point $(- 1, 1) .$
The slider on the left performs a rotation about point $A$ and the slider on the right performs a rotation about point $B .$
Perform rotations such that the sum of their angles of rotation equals $36 0^{\circ} .$
Compare the preimage and the image.

Applet to perform two rotations to a triangle

As might be checked, if the angles of rotation add up to

36 0^{\circ},

the final image is not a rotation of the original preimage but a translation.

Closure

Rotations in Real Life

In real life, there are plenty of situations where rotations can be appreciated. For instance, take a look at a door.

To open a door, the handle must be rotated.
To lock or unlock it, a key must be inserted and rotated.
While the door is opening, it rotates around the hinges.

Other examples can be found in the human body, such as joints. What happens when elbows move? They act as a center of rotation, allowing the forearm to be moved. Before moving on from this lesson, take a look around and identify some other rotations!

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