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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

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?

Watch This

http://www.youtube.com/watch?v=YkQQBH21GiQ Geometry: Rotations

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?

Answers:

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.

Vocabulary

Center of Rotation

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

Image

The image is the final appearance of a figure after a transformation operation.
Origin

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

Preimage

The pre-image is the original appearance of a figure in a transformation operation.
Rotation

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

Rigid Transformation

A rigid transformation is a transformation that preserves distance and angles, it does not change the size or shape of the figure.

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