12.4: Rotations
What if you wanted to find the center of rotation and angle of rotation for the arrows in the international recycling symbol 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. After completing this Concept, you'll be able to answer these questions.
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CK-12 Foundation: Chapter12RotationsA
Guidance
A transformation is an operation that moves, flips, or changes a figure to create a new figure. A rigid transformation is a transformation that preserves size and shape. The rigid transformations are: translations, reflections, and rotations (discussed here). The new figure created by a transformation is called the image. The original figure is called the preimage. Another word for a rigid transformation is an isometry. Rigid transformations are also called congruence transformations. If the preimage is \begin{align*}A\end{align*}
A rotation is a transformation by which a figure is turned around a fixed point to create an image. The center of rotation is 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 Concept, 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: Drawing a Rotation of \begin{align*}100^\circ\end{align*}
Tools Needed: pencil, paper, protractor, ruler
- Draw \begin{align*}\triangle ABC\end{align*} and a point \begin{align*}R\end{align*} outside the circle.
- Draw the line segment \begin{align*}\overline{RB}\end{align*}.
- Take your protractor, place the center on \begin{align*}R\end{align*} and the initial side on \begin{align*}\overline{RB}\end{align*}. Mark a \begin{align*}100^\circ\end{align*} angle.
- Find \begin{align*}B'\end{align*} such that \begin{align*}RB = RB'\end{align*}.
- Repeat steps 2-4 with points \begin{align*}A\end{align*} and \begin{align*}C\end{align*}.
- Connect \begin{align*}A', B',\end{align*} and \begin{align*}C'\end{align*} to form \begin{align*}\triangle A'B'C'\end{align*}.
This is the process you would follow to rotate any figure \begin{align*}100^\circ\end{align*} counterclockwise. If it was a different angle measure, then in Step 3, you would mark a different angle. You will need to repeat steps 2-4 for every vertex of the shape.
Common Rotations
- Rotation of \begin{align*}180^\circ\end{align*}: If \begin{align*}(x, y)\end{align*} is rotated \begin{align*}180^\circ\end{align*} around the origin, then the image will be \begin{align*}(-x, -y)\end{align*}.
- Rotation of \begin{align*}90^\circ\end{align*}: If \begin{align*}(x, y)\end{align*} is rotated \begin{align*}90^\circ\end{align*} around the origin, then the image will be \begin{align*}(-y, x)\end{align*}.
- Rotation of \begin{align*}270^\circ\end{align*}: If \begin{align*}(x, y)\end{align*} is rotated \begin{align*}270^\circ\end{align*} around the origin, then the image will be \begin{align*}(y, -x)\end{align*}.
While we can rotate any image any amount of degrees, only \begin{align*}90^\circ, 180^\circ\end{align*} and \begin{align*}270^\circ\end{align*} have special rules. To rotate a figure by an angle measure other than these three, you must use the process from the Investigation.
Example A
Rotate \begin{align*}\triangle ABC\end{align*}, with vertices \begin{align*}A(7, 4), B(6, 1),\end{align*} and \begin{align*}C(3, 1) \ 180^\circ\end{align*}. Find the coordinates of \begin{align*}\triangle A'B'C'\end{align*}.
It is very helpful to graph the triangle. If \begin{align*}A\end{align*} is \begin{align*}(7, 4)\end{align*}, that means it is 7 units to the right of the origin and 4 units up. \begin{align*}A'\end{align*} would then be 7 units to the left of the origin and 4 units down. The vertices are:
\begin{align*}A(7,4) \rightarrow A'(-7,-4)\\ B(6,1) \rightarrow B'(-6,-1)\\ C(3,1) \rightarrow C'(-3,-1)\end{align*}
Example B
Rotate \begin{align*}\overline{ST} \ 90^\circ\end{align*} counter-clockwise about the origin.
Using the \begin{align*}90^\circ\end{align*} rotation rule, \begin{align*}T'\end{align*} is (8, 2).
Example C
Find the coordinates of \begin{align*}ABCD\end{align*} after a \begin{align*}270^\circ\end{align*} rotation counter-clockwise about the origin.
Using the rule, we have:
\begin{align*}(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)\end{align*}
Watch this video for help with the Examples above.
CK-12 Foundation: Chapter12RotationsB
Concept Problem 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 \begin{align*}120^\circ\end{align*} rotation in either direction.
Vocabulary
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 new figure created by a transformation is called the image. The original figure is called the preimage. 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.
Guided Practice
1. The rotation of a quadrilateral is shown below. What is the measure of \begin{align*}x\end{align*} and \begin{align*}y\end{align*}?
2. A rotation of \begin{align*}80^\circ\end{align*} clockwise is the same as what counterclockwise rotation?
3. A rotation of \begin{align*}160^\circ\end{align*} counterclockwise is the same as what clockwise rotation?
Answers:
1. Because a rotation is an isometry that produces congruent figures, we can set up two equations to solve for \begin{align*}x\end{align*} and \begin{align*}y\end{align*}.
\begin{align*}2y &= 80^\circ && \ 2x-3 =15\\ y &= 40^\circ && \qquad 2x = 18\\ &&& \qquad \ \ x = 9\end{align*}
2. There are \begin{align*}360^\circ\end{align*} around a point. So, an \begin{align*}80^\circ\end{align*} rotation clockwise is the same as a \begin{align*}360^\circ-80^\circ=280^\circ\end{align*} rotation counterclockwise.
3. \begin{align*}360^\circ-160^\circ=200^\circ\end{align*} clockwise rotation.
Practice
In the questions below, every rotation is counterclockwise, unless otherwise stated.
- If you rotated the letter \begin{align*}p \ 180^\circ\end{align*} counterclockwise, what letter would you have?
- If you rotated the letter \begin{align*}p \ 180^\circ\end{align*} clockwise, what letter would you have? Why do you think that is?
- A \begin{align*}90^\circ\end{align*} clockwise rotation is the same as what counterclockwise rotation?
- A \begin{align*}270^\circ\end{align*} clockwise rotation is the same as what counterclockwise rotation?
- Rotating a figure \begin{align*}360^\circ\end{align*} 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.
- \begin{align*}180^\circ\end{align*}
- \begin{align*}90^\circ\end{align*}
- \begin{align*}180^\circ\end{align*}
- \begin{align*}270^\circ\end{align*}
- \begin{align*}90^\circ\end{align*}
- \begin{align*}270^\circ\end{align*}
- \begin{align*}180^\circ\end{align*}
- \begin{align*}270^\circ\end{align*}
- \begin{align*}90^\circ\end{align*}
Algebra Connection Find the measure of \begin{align*}x\end{align*} 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.
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Center of Rotation
In a rotation, the center of rotation is the point that does not move. The rest of the plane rotates around this fixed point.Image
The image is the final appearance of a figure after a transformation operation.Origin
The origin is the point of intersection of the and axes on the Cartesian plane. The coordinates of the origin are (0, 0).Preimage
The pre-image is the original appearance of a figure in a transformation operation.Rotation
A rotation is a transformation that turns a figure on the coordinate plane a certain number of degrees about a given point without changing the shape or size of the figure.Rigid Transformation
A rigid transformation is a transformation that preserves distance and angles, it does not change the size or shape of the figure.Image Attributions
Here you'll learn what a rotation is and how to find the coordinates of a rotated figure.