<meta http-equiv="refresh" content="1; url=/nojavascript/"> Rotations ( Read ) | Geometry | CK-12 Foundation
Dismiss
Skip Navigation
You are viewing an older version of this Concept. Go to the latest version.

Rotations

%
Best Score
Practice Rotations
Practice
Best Score
%
Practice Now
Rotations
 0  0  0

What if you were given the coordinates of a quadrilateral and you were asked to rotate that quadrilateral 270^\circ about the origin? What would its new coordinates be? After completing this Concept, you'll be able to rotate a figure like this one in the coordinate plane.

Watch This

Transformation: Rotation CK-12

Guidance

A transformation is an operation that moves, flips, or otherwise changes a figure to create a new figure. A rigid transformation (also known as an isometry or congruence transformation ) is a transformation that does not change the size or shape of a figure.

The rigid transformations are Transformation: Translation , Transformation: Reflection , and rotations (discussed here). The new figure created by a transformation is called the image . The original figure is called the preimage . If the preimage is A , then the image would be A' , said “a prime.” If there is an image of A' , that would be labeled A'' , said “a double prime.”

A rotation is a transformation where a figure is turned around a fixed point to create an image. The lines drawn from the preimage to the center of rotation and from the center of rotation to the image form the angle of rotation . In this concept, we will only do counterclockwise rotations .

While we can rotate any image any amount of degrees, 90^\circ, 180^\circ and 270^\circ rotations are common and have rules worth memorizing.

Rotation of 180^\circ : (x,y) \rightarrow (-x,-y)

Rotation of 90^\circ : (x,y) \rightarrow (-y,x)

Rotation of 270^\circ : (x,y) \rightarrow (y,-x)

Example A

A rotation of 80^\circ clockwise is the same as what counterclockwise rotation?

There are 360^\circ around a point. So, an 80^\circ rotation clockwise is the same as a 360^\circ-80^\circ=280^\circ rotation counterclockwise.

Example B

A rotation of 160^\circ counterclockwise is the same as what clockwise rotation?

360^\circ-160^\circ=200^\circ clockwise rotation.

Example C

Rotate \triangle ABC , with vertices A(7, 4), B(6, 1) , and C(3, 1) , 180^\circ about the origin. Find the coordinates of \triangle A'B'C' .

Use the rule above to find \triangle A'B'C' .

A(7,4) & \rightarrow A'(-7,-4)\\B(6,1) & \rightarrow B'(-6,-1)\\C(3,1) & \rightarrow C'(-3,-1)

Transformation: Rotation CK-12

Guided Practice

1. Rotate \overline{ST} \ 90^\circ .

2. Find the coordinates of ABCD after a 270^\circ rotation.

3. The rotation of a quadrilateral is shown below. What is the measure of x and y ?

Answers:

1.

2. Using the rule, we have:

(x,y) & \rightarrow (y,-x)\\ A(-4,5) & \rightarrow A'(5,4)\\B(1,2) & \rightarrow B'(2,-1)\\C(-6,-2) & \rightarrow C'(-2,6)\\D(-8,3) & \rightarrow D'(3,8)

3. Because a rotation produces congruent figures, we can set up two equations to solve for x and y .

2y &= 80^\circ && 2x-3=15\\				y &= 40^\circ && \quad \ \ 2x=18\\& && \qquad \ x = 9

Practice

In the questions below, every rotation is counterclockwise , unless otherwise stated.

  1. If you rotated the letter p \ 180^\circ counterclockwise, what letter would you have?
  2. If you rotated the letter p \ 180^\circ clockwise , what letter would you have?
  3. A 90^\circ clockwise rotation is the same as what counterclockwise rotation?
  4. A 270^\circ clockwise rotation is the same as what counterclockwise rotation?
  5. A 210^\circ counterclockwise rotation is the same as what clockwise rotation?
  6. A 120^\circ counterclockwise rotation is the same as what clockwise rotation?
  7. A 340^\circ counterclockwise rotation is the same as what clockwise rotation?
  8. Rotating a figure 360^\circ is the same as what other rotation?
  9. Does it matter if you rotate a figure 180^\circ clockwise or counterclockwise? Why or why not?
  10. When drawing a rotated figure and using your protractor, would it be easier to rotate the figure 300^\circ counterclockwise or 60^\circ clockwise? Explain your reasoning.

Rotate each figure in the coordinate plane the given angle measure. The center of rotation is the origin.

  1. 180^\circ
  2. 90^\circ
  3. 180^\circ
  4. 270^\circ
  5. 90^\circ
  6. 270^\circ
  7. 180^\circ
  8. 270^\circ
  9. 90^\circ

Find the measure of x in the rotations below. The blue figure is the preimage.

Find the angle of rotation for the graphs below. The center of rotation is the origin and the blue figure is the preimage. Your answer will be 90^\circ, 270^\circ , or 180^\circ .

Image Attributions

Reviews

Email Verified
Well done! You've successfully verified the email address .
OK
Please wait...
Please wait...

Original text