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# 12.4: Rotations

Difficulty Level: At Grade Created by: CK-12

## Learning Objectives

• Find the image of a figure in a rotation in a coordinate plane.

## Review Queue

1. Reflect XYZ\begin{align*}\triangle XYZ\end{align*} with vertices X(9,2),Y(2,4)\begin{align*}X(9, 2), Y(2, 4)\end{align*} and Z(7,8)\begin{align*}Z(7, 8)\end{align*} over the y\begin{align*}y-\end{align*}axis. What are the vertices of XYZ\begin{align*}\triangle X’Y’Z’\end{align*}?
2. Reflect XYZ\begin{align*}\triangle X’Y’Z’\end{align*} over the x\begin{align*}x-\end{align*}axis. What are the vertices of XYZ\begin{align*}\triangle X”Y”Z”\end{align*}?
3. How do the coordinates of XYZ\begin{align*}\triangle X”Y”Z”\end{align*} relate to XYZ\begin{align*}\triangle XYZ\end{align*}?

Know What? The international symbol for recycling is to the right. 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 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 section, we will only do counterclockwise rotations.

Example 1: A rotation of 80\begin{align*}80^\circ\end{align*} clockwise is the same as what counterclockwise rotation?

Solution: There are 360\begin{align*}360^\circ\end{align*} around a point. So, an 80\begin{align*}80^\circ\end{align*} rotation clockwise is the same as a 36080=280\begin{align*}360^\circ-80^\circ=280^\circ\end{align*} rotation counterclockwise.

Example 2: A rotation of 160\begin{align*}160^\circ\end{align*} counterclockwise is the same as what clockwise rotation?

Solution: 360160=200\begin{align*}360^\circ-160^\circ=200^\circ\end{align*} clockwise rotation.

Investigation 12-1: Drawing a Rotation of 100\begin{align*}100^\circ\end{align*}

Tools Needed: pencil, paper, protractor, ruler

1. Draw ABC\begin{align*}\triangle ABC\end{align*} and a point R\begin{align*}R\end{align*}.
2. Draw RB¯¯¯¯¯¯¯¯\begin{align*}\overline{RB}\end{align*}.
3. Place the center of a protractor on R\begin{align*}R\end{align*} and the 0\begin{align*}0^\circ\end{align*} line on RB¯¯¯¯¯¯¯¯\begin{align*}\overline{RB}\end{align*}. Mark a 100\begin{align*}100^\circ\end{align*} angle.
4. Mark B\begin{align*}B’\end{align*} on the 100\begin{align*}100^\circ\end{align*} line so RB=RB\begin{align*}RB = RB’\end{align*}.
5. Repeat steps 2-4 with A\begin{align*}A\end{align*} and C\begin{align*}C\end{align*}.
6. Make ABC\begin{align*}\triangle A’B’C’\end{align*}.

Use this process to rotate any figure.

Example 3: Rotate rectangle RECT\begin{align*}RECT\end{align*} 80\begin{align*}80^\circ\end{align*} counterclockwise around P\begin{align*}P\end{align*}.

Solution: Use Investigation 12-1. In step 3, change the angle to 80\begin{align*}80^\circ\end{align*}. Each angle of rotation is 80\begin{align*}80^\circ\end{align*}.

mRPRmEPEmCPCmTPT=80=80=80=80\begin{align*}m \angle RPR’ &= 80^\circ\\ m \angle EPE’ &= 80^\circ\\ m \angle CPC’ &= 80^\circ\\ m \angle TPT’ &= 80^\circ\end{align*}

180\begin{align*}180^\circ\end{align*} Rotation

To rotate a figure 180\begin{align*}180^\circ\end{align*}, in the xy\begin{align*}x-y\end{align*} plane, we use the origin as the center of the rotation. A 180\begin{align*}180^\circ\end{align*} angle is called a straight angle. So, an image rotated over the origin 180\begin{align*}180^\circ\end{align*} will be on the same line and the same distance away from the origin as the preimage, but on the other side.

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

Solution: You can either use Investigation 12-1 or the hint given above to find ABC\begin{align*}\triangle A’B’C’\end{align*}. First, graph the triangle. If A\begin{align*}A\end{align*} is (7, 4), that means it is 7 units to the right of the origin and 4 units up. A\begin{align*}A’\end{align*} would then be 7 units to the left of the origin and 4 units down.

A(7,4)B(6,1)C(3,1)A(7,4)B(6,1)C(3,1)\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*} Rotation of 180\begin{align*}180^\circ\end{align*}: (x,y)(x,y)\begin{align*}(x,y) \rightarrow (-x,-y)\end{align*}

Recall from the second section that a rotation is an isometry. This means that ABCABC\begin{align*}\triangle ABC \cong \triangle A’B’C’\end{align*}. You can use the distance formula to show this.

90\begin{align*}90^\circ\end{align*} Rotation

Similar to the 180\begin{align*}180^\circ\end{align*} rotation, the image of a 90\begin{align*}90^\circ\end{align*} will be the same distance away from the origin as its preimage, but rotated 90\begin{align*}90^\circ\end{align*}.

Example 5: Rotate ST¯¯¯¯¯¯¯ 90\begin{align*}\overline{ST} \ 90^\circ\end{align*}.

Solution: When rotating something 90\begin{align*}90^\circ\end{align*}, use Investigation 12-1 to see if there is a pattern.

Rotation of 90\begin{align*}90^\circ\end{align*}: (x,y)(y,x)\begin{align*}(x,y) \rightarrow (-y,x)\end{align*}

Rotation of 270\begin{align*}270^\circ\end{align*}

A rotation of 270\begin{align*}270^\circ\end{align*} counterclockwise would be the same as a rotation of 90\begin{align*}90^\circ\end{align*} plus a rotation of 180\begin{align*}180^\circ\end{align*}. So, if the values of a 90\begin{align*}90^\circ\end{align*} rotation are (y,x)\begin{align*}(-y, x)\end{align*}, then a 270\begin{align*}270^\circ\end{align*} rotation would be the opposite sign of each, or (y,x)\begin{align*}(y, -x)\end{align*}.

Rotation of 270\begin{align*}270^\circ\end{align*}: (x,y)(y,x)\begin{align*}(x,y) \rightarrow (y,-x)\end{align*}

Example 6: Find the coordinates of ABCD\begin{align*}ABCD\end{align*} after a 270\begin{align*}270^\circ\end{align*} rotation.

Solution: Using the rule, we have:

(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)\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*}

While we can rotate any image any amount of degrees, only 90,180\begin{align*}90^\circ, 180^\circ\end{align*} and 270\begin{align*}270^\circ\end{align*} have special rules. To rotate a figure by an angle measure other than these three, you must use Investigation 12-1.

Example 7: Algebra Connection The rotation of a quadrilateral is shown below. What is the measure of x\begin{align*}x\end{align*} and y\begin{align*}y\end{align*}?

Solution: Because a rotation is an isometry, we can set up two equations to solve for x\begin{align*}x\end{align*} and y\begin{align*}y\end{align*}.

2yy=80=402x3=15  2x=18 x=9\begin{align*}2y &= 80^\circ && 2x-3=15\\ y &= 40^\circ && \quad \ \ 2x=18\\ & && \qquad \ x = 9\end{align*}

Know What? Revisited The center of rotation is shown in the picture to the right. If we draw rays to the same place in each arrow, the two images are a 120\begin{align*}120^\circ\end{align*} rotation in either direction.

## Review Questions

• Questions 1-10 are similar to Examples 1 and 2.
• Questions 11-16 are similar to Investigation 12-1 and Example 3.
• Questions 17-25 are similar to Examples 4-6.
• Questions 26-28 are similar to Example 7.
• Questions 29-34 are similar to Examples 4-6.
• Questions 34-37 are a review.
• Question 38 is similar to Example 4.

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

1. If you rotated the letter p 180\begin{align*}p \ 180^\circ\end{align*} counterclockwise, what letter would you have?
2. If you rotated the letter p 180\begin{align*}p \ 180^\circ\end{align*} clockwise, what letter would you have?
3. A 90\begin{align*}90^\circ\end{align*} clockwise rotation is the same as what counterclockwise rotation?
4. A 270\begin{align*}270^\circ\end{align*} clockwise rotation is the same as what counterclockwise rotation?
5. A 210\begin{align*}210^\circ\end{align*} counterclockwise rotation is the same as what clockwise rotation?
6. A 120\begin{align*}120^\circ\end{align*} counterclockwise rotation is the same as what clockwise rotation?
7. A 340\begin{align*}340^\circ\end{align*} counterclockwise rotation is the same as what clockwise rotation?
8. Rotating a figure 360\begin{align*}360^\circ\end{align*} is the same as what other rotation?
9. Does it matter if you rotate a figure 180\begin{align*}180^\circ\end{align*} 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\begin{align*}300^\circ\end{align*} counterclockwise or 60\begin{align*}60^\circ\end{align*} clockwise? Explain your reasoning.

Using Investigation 12-1, rotate each figure around point P\begin{align*}P\end{align*} the given angle measure.

1. 50\begin{align*}50^\circ\end{align*}
2. 120\begin{align*}120^\circ\end{align*}
3. 200\begin{align*}200^\circ\end{align*}
4. 330\begin{align*}330^\circ\end{align*}
5. 75\begin{align*}75^\circ\end{align*}
6. 170\begin{align*}170^\circ\end{align*}

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

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

Algebra Connection Find the measure of x\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. Your answer will be 90,270\begin{align*}90^\circ, 270^\circ\end{align*}, or 180\begin{align*}180^\circ\end{align*}.

Two Reflections The vertices of GHI\begin{align*}\triangle GHI\end{align*} are G(2,2),H(8,2)\begin{align*}G(-2, 2), H(8, 2)\end{align*}, and I(6,8)\begin{align*}I(6, 8)\end{align*}. Use this information to answer questions 24-27.

1. Plot GHI\begin{align*}\triangle GHI\end{align*} on the coordinate plane.
2. Reflect GHI\begin{align*}\triangle GHI\end{align*} over the x\begin{align*}x-\end{align*}axis. Find the coordinates of GHI\begin{align*}\triangle G’H’I’\end{align*}.
3. Reflect GHI\begin{align*}\triangle G’H’I’\end{align*} over the y\begin{align*}y-\end{align*}axis. Find the coordinates of GHI\begin{align*}\triangle G”H”I”\end{align*}.
4. What one transformation would be the same as this double reflection?

## Review Queue Answers

1. X(9,2),Y(2,4),Z(7,8)\begin{align*}X'(-9, 2), Y'(-2, 4), Z'(-7, 8)\end{align*}
2. \begin{align*}X''(-9, -2), Y''(-2, -4), Z''(-7, -8)\end{align*}
3. \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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