You want your friend to rotate the pentagon \begin{align*}90^\circ\end{align*}

#### Guidance

A **rotation** is one example of a **rigid transformation**. A rotation of \begin{align*}t^\circ\end{align*}

Keep in mind that the location of point \begin{align*}O\end{align*}

There is a circle with center \begin{align*}O\end{align*}

And here is that connection for \begin{align*}D\end{align*} and \begin{align*}D^\prime\end{align*}.

When describing a rotation, it is important to consider three pieces of information. First, the **center of rotation**. This is the point about which each point turns (the center of each circle passing through corresponding points). Second, the **direction of rotation**. You can rotate clockwise or counterclockwise. Third, the degree of rotation. This tells you how much you are turning. Remember that a full circle \begin{align*}360^\circ\end{align*}.

**Example A**

Describe the rotation below.

**Solution:** This is a \begin{align*}135^\circ\end{align*} counterclockwise rotation about point \begin{align*}O\end{align*}. You could also say this is a \begin{align*}225^\circ\end{align*} clockwise rotation about point \begin{align*}O\end{align*}. Once the rotation has taken place, there is no way to know the direction of rotation.

**Example B**

Is the following transformation a rotation?

**Solution:** No, while \begin{align*}\angle DOD^\prime\end{align*} is \begin{align*}180^\circ\end{align*}, the angles connecting other pairs of points and point \begin{align*}O\end{align*} are not \begin{align*}180^\circ\end{align*}.

This transformation is a reflection across a line through point \begin{align*}O\end{align*}, perpendicular to \begin{align*}\overline{DD^\prime}\end{align*}.

**Example C**

Rotate the rectangle \begin{align*}90^\circ\end{align*} counterclockwise about the origin. How are the points of the original rectangle connected to the points of the image?

**Solution:** Pick one point, such as point \begin{align*}A\end{align*}, and draw a line segment connecting it to the origin. Because perpendicular lines form \begin{align*}90^\circ\end{align*} angles, to rotate \begin{align*}90^\circ\end{align*}, find a line perpendicular to the line segment connecting \begin{align*}A\end{align*} to the origin. *Remember that perpendicular lines have opposite reciprocal slopes.* \begin{align*}A^\prime\end{align*} will be on the perpendicular line, the same distance from the origin as \begin{align*}A\end{align*} was.

Once you have \begin{align*}A^\prime\end{align*}, you can build the rectangle around it. \begin{align*}A\end{align*} was at point (-3, 1) and \begin{align*}A^\prime\end{align*} is at point (-1, -3). In general, a \begin{align*}90^\circ\end{align*} counterclockwise rotation about the origin takes \begin{align*}(x,y)\end{align*} and maps it to \begin{align*}(-y,x)\end{align*}.

**Concept Problem Revisited**

To make sure your friend correctly rotates the pentagon \begin{align*}90^\circ\end{align*}, you need to also tell her the **center of rotation** and the **direction** **of rotation.**

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#### Vocabulary

A *rotation of \begin{align*}t^\circ\end{align*}*** about a given point **\begin{align*}O\end{align*} takes each point on a shape \begin{align*}P\end{align*} and moves it to \begin{align*}P^\prime\end{align*} such that \begin{align*}P^\prime\end{align*} is on the circle with center \begin{align*}O\end{align*} and radius \begin{align*}\overline{OP}\end{align*} and \begin{align*}\angle P^\prime OP=t^\circ\end{align*}. Point \begin{align*}O\end{align*} is the

*center of rotation.*
A ** rigid transformation** is a transformation that preserves distance and angles.

#### Guided Practice

1. Describe the rotation below.

2. Rotate the rectangle \begin{align*}180^\circ\end{align*} counterclockwise about the origin. How do the points of the original rectangle compare to their corresponding points on the image?

3. How would your answer to #2 be different if instead you rotated \begin{align*}180^\circ\end{align*} clockwise about the origin?

**Answers:**

1. This is a \begin{align*}135^\circ\end{align*} clockwise rotation about point \begin{align*}B\end{align*}. You could also say this is a \begin{align*}225^\circ\end{align*} counterclockwise rotation about point \begin{align*}B\end{align*}.

2. You can see that there is a straight line \begin{align*}(180^\circ)\end{align*} passing through the origin connecting each pair of corresponding points. \begin{align*}(x, y) \rightarrow (-x, -y)\end{align*}.

3. There would be no difference. A rotation \begin{align*}180^\circ\end{align*} clockwise is the same as a rotation \begin{align*}180^\circ\end{align*} counterclockwise.

#### Practice

Describe each of the following rotations in TWO ways.

1.

2.

3.

4. What does a \begin{align*}360^\circ\end{align*} rotation look like?

5. Rotate the rectangle below \begin{align*}90^\circ\end{align*} clockwise about the origin.

6. Use the previous example to help you describe what happens to a point \begin{align*}(x, y)\end{align*} that is rotated \begin{align*}90^\circ\end{align*} clockwise about the origin.

7. (3, 2) is rotated \begin{align*}90^\circ\end{align*} clockwise about the origin. Where does it end up?

8. (3, 2) is rotated \begin{align*}180^\circ\end{align*} clockwise about the origin. Where does it end up? *Hint: Use Guided Practice #2 for help.*

9. (3, 2) is rotated \begin{align*}90^\circ\end{align*} counterclockwise about the origin. Where does it end up? *Hint: Use Example C.*

10. What do circles have to do with rotations?

11. \begin{align*}\Delta ABC\end{align*} is rotated \begin{align*}130^\circ\end{align*} clockwise about point \begin{align*}O\end{align*} to create \begin{align*}\Delta A^\prime B^\prime C^\prime\end{align*}. Describe two connections between the points \begin{align*}B, O\end{align*}, and \begin{align*}B^\prime\end{align*}.

For 12-15, consider \begin{align*}\Delta ABC\end{align*} below.

12. Rotate \begin{align*}\Delta ABC \ 90^\circ\end{align*} counterclockwise about point \begin{align*}B\end{align*}.

13. Rotate \begin{align*}\Delta ABC \ 90^\circ\end{align*} counterclockwise about point \begin{align*}A\end{align*}.

14. Rotate \begin{align*}\Delta ABC \ 90^\circ\end{align*} counterclockwise about point \begin{align*}C\end{align*}.

15. Could you have used the rule that a \begin{align*}90^\circ\end{align*} counterclockwise rotation takes \begin{align*}(x, y)\end{align*} to \begin{align*}(-y, x)\end{align*} to perform the previous three rotations? Explain.