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Congruence, Proof, and Constructions

Rotations of Figures in a Plane

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 and its image under a rotation.

Catch-Up and Review

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

Explore

Rotating Points About a Center

In the following applet, points and can be moved and rotated about point
Rotating Points About a Point
Make an arrangement of points and rotate them about Then, find the distance between and each preimage, and the distance between and each image. 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 about point Use a protractor to find the measure of and
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 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 by a certain angle measure The 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 the same distance from and has a measure of
The angle formed by a preimage, the center of rotation, and the image is called the angle of rotation and its measure is 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 Use the given measuring tool to find the distance from each vertex to and the angles formed by each preimage, the point and the corresponding image.
Image of a Triangle After a Rotation
Which of the triangles, or , is the image of under a rotation about
If one of the triangles is a rotation of about 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
Image of a Triangle After a Rotation
Notice that the vertices of are further from than the vertices of Consequently, cannot be the image of under a rotation about Next, find and compare the measures of and
Image of a Triangle After a Rotation
As can be seen, and have all the same measure, which is when measured counterclockwise or when measured clockwise. Therefore, is the image of under a rotation about The angle of rotation is either or

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

1

Draw
Using a straightedge, draw the segment connecting the center of rotation and point
Drawing the segment connecting P and A

2

Place the Protractor

Place the center of the protractor on and align it with

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

Making a mark at 130 degrees

4

Draw a Ray
Using the straightedge, draw a ray with starting point 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
Place the compass tip on and open it to the distance between and With this setting, make an arc centered at 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 under the give rotation. Notice that, by construction, is equal to
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 counterclockwise rotation to about point

Rotating a Triangle About a Point

Draw and its image under this rotation.

Answer

Rotating a Triangle About a Point

Hint

Rotate the vertices of one at a time. Then, draw the triangle formed by the three images.

Solution

To rotate , the rotation can be performed on each vertex, one at a time. For example, start by rotating To do so, first draw using a straightedge.
Rotating a Triangle About a Point
Then, place the center of the protractor on and align it with in such a way that the rotation is counterclockwise. Then, make a small mark at
Rotating a Triangle About a Point
Next, draw a ray with starting point that passes through the mark made before.
Rotating a Triangle About a Point
Finally, place the compass tip on and open the compass to the distance between and With this setting, make an arc that intersects the ray. The intersection point is the image of under the rotation.
Rotating a Triangle About a Point
Vertices and can be rotated following the same steps.
Rotating a Triangle About a Point

Finally, the image of under the given rotation is the triangle formed by and

Rotating a Triangle About a Point

Pop Quiz

Practice Rotations

Rotate about point by the specified angle measure. To do so, place points and where they should be after the rotation. Use the measuring tool if needed.

Performing random rotations to random triangles

Discussion

Rotating Points by 90°, 180°, and 270°

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 or 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 under a rotation about the origin is
  • The image of under a rotation about the origin is
  • The image of under a rotation about the origin is

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 is the image of under a certain rotation.

Preimage of a Quadrilateral After a Rotation

Find the center and angle of rotation.

Answer

Angle of rotation: clockwise or counterclockwise.
Graph:

Preimage of a Quadrilateral After a Rotation

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 for instance. Then, with the aid of a compass and a straightedge, start by constructing the perpendicular bisector of this segment.

Preimage of a Quadrilateral After a Rotation

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

Preimage of a Quadrilateral After a 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.

Preimage of a Quadrilateral After a Rotation

The angle of rotation is either counterclockwise or clockwise.

Pop Quiz

Practice Finding Angles of Rotations

In the following applet, the solid triangle is the image of the dashed triangle under a certain counterclockwise rotation. Move the point to find the center and write the angle of rotation. The possible angles of rotation are or

Performing random rotations to random triangles

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.

  • counterclockwise rotation about
  • counterclockwise rotation about

LaShay has to perform both rotations to 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 are Knowing this, help LaShay to find the coordinates of and draw

Answer

Triangles After Two Different Rotations

Hint

Recall that a counterclockwise rotation about the origin maps point onto

Solution

Start by applying any of the given two rotations. For example, apply first. Notice it is a counterclockwise rotation about the origin. Then, it maps onto Knowing this, the image of and can be quickly found. Therefore, the image of under looks as follows.

Rotating a Triangle About a Point
Next, apply to Remember, it is a counterclockwise rotation about
Rotating a Triangle About a Point
Since the coordinates of are the same as the book says, the rotations were applied in the correct order. Consequently, the coordinates of are Notice that applying the rotations in the reverse order would produce a different image.
Rotating a Triangle About a Point

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 onto performing only one rotation? The answer is yes! A counterclockwise rotation about 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 Investigate what happens in this case by using the following applet.
  • Place the centers and at any place within the square formed by the axes and the point
  • The slider on the left performs a rotation about point and the slider on the right performs a rotation about point
  • Perform rotations such that the sum of their angles of rotation equals
  • 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 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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