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Rotations

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You want your friend to rotate the pentagon $90^\circ$ . What are the two other pieces of information you need to tell your friend to ensure that she rotates the pentagon the way you are imagining?

Guidance

A rotation is one example of a rigid transformation . A rotation of  $t^\circ$ about a given point  $O$ takes each point on a shape  $P$ and moves it to  $P^\prime$ such that  $P^\prime$ is on the circle with center  $O$ and radius $\overline{OP}$  and $\angle P^\prime OP=t^\circ$ . Below, the parallelogram has been rotated  $90^\circ$ counterclockwise about point  $O$ to create a new parallelogram (the image).

Keep in mind that the location of point  $O$ matters. The parallelogram below was also rotated $90^\circ$  counterclockwise about point $O$ , but point  $O$ is in a different location.

There is a circle with center  $O$ that passes through each pair of corresponding points. You can see the angles are  $90^\circ$ between each pair of corresponding points through point $O$ . Below, you can see this connection for  $C$ and $C^\prime$ .

And here is that connection for  $D$ and $D^\prime$ .

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 $360^\circ$ .

Example A

Describe the rotation below.

Solution: This is a $135^\circ$  counterclockwise rotation about point $O$ . You could also say this is a $225^\circ$  clockwise rotation about point $O$ . 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 $\angle DOD^\prime$  is $180^\circ$ , the angles connecting other pairs of points and point  $O$ are not $180^\circ$ .

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

Example C

Rotate the rectangle $90^\circ$  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 $A$ , and draw a line segment connecting it to the origin. Because perpendicular lines form $90^\circ$  angles, to rotate $90^\circ$ , find a line perpendicular to the line segment connecting  $A$ to the origin. Remember that perpendicular lines have opposite reciprocal slopes. $A^\prime$ will be on the perpendicular line, the same distance from the origin as  $A$ was.

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

Concept Problem Revisited

To make sure your friend correctly rotates the pentagon $90^\circ$ , you need to also tell her the center of rotation and the direction of rotation.

Vocabulary

A rotation of  $t^\circ$ about a given point  $O$ takes each point on a shape  $P$ and moves it to  $P^\prime$ such that  $P^\prime$ is on the circle with center  $O$ and radius $\overline{OP}$  and $\angle P^\prime OP=t^\circ$ . Point  $O$ 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 $180^\circ$  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 $180^\circ$  clockwise about the origin?

1. This is a $135^\circ$  clockwise rotation about point $B$ . You could also say this is a $225^\circ$  counterclockwise rotation about point $B$ .

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

3. There would be no difference. A rotation $180^\circ$  clockwise is the same as a rotation  $180^\circ$ counterclockwise.

Practice

Describe each of the following rotations in TWO ways.

1.

2.

3.

4. What does a $360^\circ$  rotation look like?

5. Rotate the rectangle below $90^\circ$  clockwise about the origin.

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

7. (3, 2) is rotated $90^\circ$  clockwise about the origin. Where does it end up?

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

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

10. What do circles have to do with rotations?

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

For 12-15, consider  $\Delta ABC$ below.

12. Rotate  $\Delta ABC \ 90^\circ$ counterclockwise about point $B$ .

13. Rotate  $\Delta ABC \ 90^\circ$ counterclockwise about point $A$ .

14. Rotate  $\Delta ABC \ 90^\circ$ counterclockwise about point $C$ .

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