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Rotations

Transformations by which a figure is turned around a fixed point to create an image.

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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

Rotations 

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, then the image would be labeled \begin{align*}A'\end{align*}A, said “a prime.” If there is an image of \begin{align*}A'\end{align*}A, that would be labeled \begin{align*}A''\end{align*}A′′, said “a double prime.”

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 Lesson, 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*}100

Tools Needed: pencil, paper, protractor, ruler

  1. Draw \begin{align*}\triangle ABC\end{align*}ABC and a point \begin{align*}R\end{align*}R outside the circle.
  2. Draw the line segment \begin{align*}\overline{RB}\end{align*}RB¯¯¯¯¯¯¯¯.
  3. Take your protractor, place the center on \begin{align*}R\end{align*}R and the initial side on \begin{align*}\overline{RB}\end{align*}RB¯¯¯¯¯¯¯¯. Mark a \begin{align*}100^\circ\end{align*}100 angle.
  4. Find \begin{align*}B'\end{align*}B such that \begin{align*}RB = RB'\end{align*}RB=RB.
  5. Repeat steps 2-4 with points \begin{align*}A\end{align*}A and \begin{align*}C\end{align*}C.
  6. Connect \begin{align*}A', B',\end{align*}A,B, and \begin{align*}C'\end{align*}C to form \begin{align*}\triangle A'B'C'\end{align*}ABC.

This is the process you would follow to rotate any figure \begin{align*}100^\circ\end{align*}100 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*}180: If \begin{align*}(x, y)\end{align*}(x,y) is rotated \begin{align*}180^\circ\end{align*}180 around the origin, then the image will be \begin{align*}(-x, -y)\end{align*}(x,y).
  • Rotation of \begin{align*}90^\circ\end{align*}90: If \begin{align*}(x, y)\end{align*}(x,y) is rotated \begin{align*}90^\circ\end{align*}90 around the origin, then the image will be \begin{align*}(-y, x)\end{align*}(y,x).
  • Rotation of \begin{align*}270^\circ\end{align*}270: If \begin{align*}(x, y)\end{align*}(x,y) is rotated \begin{align*}270^\circ\end{align*}270 around the origin, then the image will be \begin{align*}(y, -x)\end{align*}(y,x).

While we can rotate any image any amount of degrees, only \begin{align*}90^\circ, 180^\circ\end{align*}90,180 and \begin{align*}270^\circ\end{align*}270 have special rules. To rotate a figure by an angle measure other than these three, you must use the process from the Investigation.

 

 

Rotating a Triangle 

Rotate \begin{align*}\triangle ABC\end{align*}ABC, with vertices \begin{align*}A(7, 4), B(6, 1),\end{align*}A(7,4),B(6,1), and \begin{align*}C(3, 1) \ 180^\circ\end{align*}C(3,1) 180. Find the coordinates of \begin{align*}\triangle A'B'C'\end{align*}ABC.

It is very helpful to graph the triangle. If \begin{align*}A\end{align*}A is \begin{align*}(7, 4)\end{align*}(7,4), that means it is 7 units to the right of the origin and 4 units up. \begin{align*}A'\end{align*}A 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*}A(7,4)A(7,4)B(6,1)B(6,1)C(3,1)C(3,1)

Rotating a Line Segment 

Rotate \begin{align*}\overline{ST} \ 90^\circ\end{align*}ST¯¯¯¯¯¯¯ 90 counter-clockwise about the origin.

Using the \begin{align*}90^\circ\end{align*}90 rotation rule, \begin{align*}T'\end{align*}T is (8, 2).

Finding Coordinates After a Rotation 

Find the coordinates of \begin{align*}ABCD\end{align*}ABCD after a \begin{align*}270^\circ\end{align*}270 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*}(x,y)A(4,5)B(1,2)C(6,2)D(8,3)(y,x)A(5,4)B(2,1)C(2,6)D(3,8)

 

 

Recycling 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.

Examples

Example 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*}?

 

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*}

Example 2

A rotation of \begin{align*}80^\circ\end{align*} clockwise is the same as what counterclockwise rotation?

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.

 

Example 3

A rotation of \begin{align*}160^\circ\end{align*} counterclockwise is the same as what clockwise rotation?

\begin{align*}360^\circ-160^\circ=200^\circ\end{align*} clockwise rotation.

Review 

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

  1. If you rotated the letter \begin{align*}p \ 180^\circ\end{align*} counterclockwise, what letter would you have?
  2. If you rotated the letter \begin{align*}p \ 180^\circ\end{align*} clockwise, what letter would you have? Why do you think that is?
  3. A \begin{align*}90^\circ\end{align*} clockwise rotation is the same as what counterclockwise rotation?
  4. A \begin{align*}270^\circ\end{align*} clockwise rotation is the same as what counterclockwise rotation?
  5. 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.

  1. \begin{align*}180^\circ\end{align*}
  2. \begin{align*}90^\circ\end{align*}
  3. \begin{align*}180^\circ\end{align*}
  4. \begin{align*}270^\circ\end{align*}
  5. \begin{align*}90^\circ\end{align*}
  6. \begin{align*}270^\circ\end{align*}
  7. \begin{align*}180^\circ\end{align*}
  8. \begin{align*}270^\circ\end{align*}
  9. \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.

Review (Answers)

To view the Review answers, open this PDF file and look for section 12.4. 

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Vocabulary

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 x and y 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

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