12.4: Rotations
Learning Objectives
 Find the image of a figure in a rotation in a coordinate plane.
 Recognize that a rotation is an isometry.
Review Queue
 Reflect
△XYZ with verticesX(9,2),Y(2,4) andZ(7,8) over they− axis. What are the vertices of△X′Y′Z′ ?  Reflect
△X′Y′Z′ over thex− axis. What are the vertices of△X′′Y′′Z′′ ?  How do the coordinates of
△X′′Y′′Z′′ relate to△XYZ ?
Know What? The international symbol for recycling appears below. It is three arrows rotated around a point. Let’s assume that the arrow on the top is the preimage and the other two are its images. Find the center of rotation and the angle of rotation for each image.
Defining Rotations
Rotation: A transformation by which a figure is turned around a fixed point to create an image.
Center of Rotation: The fixed point that a figure is rotated around.
Lines can be drawn from the preimage to the center of rotation, and from the center of rotation to the image. The angle formed by these lines is the angle of rotation.
In this section, our center of rotation will always be the origin. Rotations can also be clockwise or counterclockwise. We will only do counterclockwise rotations, to go along with the way the quadrants are numbered.
Investigation 121: Drawing a Rotation of
Tools Needed: pencil, paper, protractor, ruler
 Draw
△ABC and a pointR outside the circle.  Draw the line segment
RB¯¯¯¯¯¯¯¯ .  Take your protractor, place the center on
R and the initial side onRB¯¯¯¯¯¯¯¯ . Mark a100∘ angle.  Find
B′ such thatRB=RB′ .  Repeat steps 24 with points
A andC .  Connect
A′,B′, andC′ to form△A′B′C′ .
This is the process you would follow to rotate any figure
To rotate a figure
Example 1: Rotate
Solution: You can either use Investigation 121 or the hint given above to find
The image has vertices that are the negative of the preimage. This will happen every time a figure is rotated
Rotation of
From this example, we can also see that a rotation is an isometry. This means that
Similar to the
Example 2: Rotate
Solution: When we rotate something
Using this pattern,
If you were to write the slope of each point to the origin,
Rotation of
Rotation of
A rotation of
Rotation of
Example 3: Find the coordinates of
Solution: Using the rule, we have:
While we can rotate any image any amount of degrees, only
Example 4: Algebra Connection The rotation of a quadrilateral is shown below. What is the measure of
Solution: Because a rotation is an isometry, we can set up two equations to solve for
Know What? Revisited The center of rotation is shown in the picture below. If we draw rays to the same point in each arrow, we see that the two images are a
Review Questions
In the questions below, every rotation is counterclockwise, unless otherwise stated.
Using Investigation 121, rotate each figure around point

50∘ 
120∘ 
200∘  If you rotated the letter
p 180∘ counterclockwise, what letter would you have?  If you rotated the letter
p 180∘ clockwise, what letter would you have? Why do you think that is?  A
90∘ clockwise rotation is the same as what counterclockwise rotation?  A
270∘ clockwise rotation is the same as what counterclockwise rotation?  Rotating a figure
360∘ is the same as what other rotation?
Rotate each figure in the coordinate plane the given angle measure. The center of rotation is the origin.

180∘ 
90∘ 
180∘ 
270∘ 
90∘ 
270∘ 
180∘ 
270∘ 
90∘
Algebra Connection Find the measure of
Find the angle of rotation for the graphs below. The center of rotation is the origin and the blue figure is the preimage.
Two Reflections The vertices of
 Plot
△GHI on the coordinate plane.  Reflect
△GHI over thex− axis. Find the coordinates of△G′H′I′ .  Reflect
△G′H′I′ over they− axis. Find the coordinates of△G′′H′′I′′ .  What one transformation would be the same as this double reflection?
Multistep Construction Problem
 Draw two lines that intersect, \begin{align*}m\end{align*} and \begin{align*}n\end{align*}, and \begin{align*}\triangle ABC\end{align*}. Reflect \begin{align*}\triangle ABC\end{align*} over line \begin{align*}m\end{align*} to make \begin{align*}\triangle A'B'C'\end{align*}. Reflect \begin{align*}\triangle A'B'C'\end{align*} over line \begin{align*}n\end{align*} to get \begin{align*}\triangle A''B''C''\end{align*}. Make sure \begin{align*}\triangle ABC\end{align*} does not intersect either line.
 Draw segments from the intersection point of lines \begin{align*}m\end{align*} and \begin{align*}n\end{align*} to \begin{align*}A\end{align*} and \begin{align*}A''\end{align*}. Measure the angle between these segments. This is the angle of rotation between \begin{align*}\triangle ABC\end{align*} and \begin{align*}\triangle A''B''C''\end{align*}.
 Measure the angle between lines \begin{align*}m\end{align*} and \begin{align*}n\end{align*}. Make sure it is the angle which contains \begin{align*}\triangle A'B'C'\end{align*} in the interior of the angle.
 What is the relationship between the angle of rotation and the angle between the two lines of reflection?
Review Queue Answers
 \begin{align*}X'(9, 2), Y'(2, 4), Z'(7, 8)\end{align*}
 \begin{align*}X''(9, 2), Y''(2, 4), Z''(7, 8)\end{align*}
 \begin{align*}\triangle X''Y''Z''\end{align*} is the double negative of \begin{align*}\triangle XYZ; (x, y) \rightarrow (x, y)\end{align*}
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