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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 \begin{align*}\triangle XYZ\end{align*} with vertices \begin{align*}X(9, 2), Y(2, 4)\end{align*} and \begin{align*}Z(7, 8)\end{align*} over the \begin{align*}y-\end{align*}axis. What are the vertices of \begin{align*}\triangle X’Y’Z’\end{align*}?
2. Reflect \begin{align*}\triangle X’Y’Z’\end{align*} over the \begin{align*}x-\end{align*}axis. What are the vertices of \begin{align*}\triangle X”Y”Z”\end{align*}?
3. How do the coordinates of \begin{align*}\triangle X”Y”Z”\end{align*} relate to \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 \begin{align*}80^\circ\end{align*} clockwise is the same as what counterclockwise rotation?

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

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

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

Tools Needed: pencil, paper, protractor, ruler

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

Use this process to rotate any figure.

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

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

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

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

To rotate a figure \begin{align*}180^\circ\end{align*}, in the \begin{align*}x-y\end{align*} plane, we use the origin as the center of the rotation. A \begin{align*}180^\circ\end{align*} angle is called a straight angle. So, an image rotated over the origin \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 \begin{align*}\triangle ABC\end{align*}, with vertices \begin{align*}A(7, 4), B(6, 1)\end{align*}, and \begin{align*}C(3, 1)\end{align*}, \begin{align*}180^\circ\end{align*}. Find the coordinates of \begin{align*}\triangle A’B’C’\end{align*}.

Solution: You can either use Investigation 12-1 or the hint given above to find \begin{align*}\triangle A’B’C’\end{align*}. First, graph the triangle. If \begin{align*}A\end{align*} is (7, 4), 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.

\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 \begin{align*}180^\circ\end{align*}: \begin{align*}(x,y) \rightarrow (-x,-y)\end{align*}

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

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

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

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

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

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

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

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

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

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

Solution: 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*}

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 Investigation 12-1.

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

Solution: Because a rotation is an isometry, 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 && \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 \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 \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?
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. A \begin{align*}210^\circ\end{align*} counterclockwise rotation is the same as what clockwise rotation?
6. A \begin{align*}120^\circ\end{align*} counterclockwise rotation is the same as what clockwise rotation?
7. A \begin{align*}340^\circ\end{align*} counterclockwise rotation is the same as what clockwise rotation?
8. Rotating a figure \begin{align*}360^\circ\end{align*} is the same as what other rotation?
9. Does it matter if you rotate a figure \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 \begin{align*}300^\circ\end{align*} counterclockwise or \begin{align*}60^\circ\end{align*} clockwise? Explain your reasoning.

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

1. \begin{align*}50^\circ\end{align*}
2. \begin{align*}120^\circ\end{align*}
3. \begin{align*}200^\circ\end{align*}
4. \begin{align*}330^\circ\end{align*}
5. \begin{align*}75^\circ\end{align*}
6. \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. \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. Your answer will be \begin{align*}90^\circ, 270^\circ\end{align*}, or \begin{align*}180^\circ\end{align*}.

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

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

1. \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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