You want your friend to rotate the pentagon . 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?
http://www.youtube.com/watch?v=YkQQBH21GiQ Geometry: Rotations
A rotation is one example of a rigid transformation . A rotation of about a given point takes each point on a shape and moves it to such that is on the circle with center and radius and . Below, the parallelogram has been rotated counterclockwise about point to create a new parallelogram (the image).
Keep in mind that the location of point matters. The parallelogram below was also rotated counterclockwise about point , but point is in a different location.
There is a circle with center that passes through each pair of corresponding points. You can see the angles are between each pair of corresponding points through point . Below, you can see this connection for and .
And here is that connection for and .
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 .
Describe the rotation below.
Solution: This is a counterclockwise rotation about point . You could also say this is a clockwise rotation about point . Once the rotation has taken place, there is no way to know the direction of rotation.
Is the following transformation a rotation?
Solution: No, while is , the angles connecting other pairs of points and point are not .
This transformation is a reflection across a line through point , perpendicular to .
Rotate the rectangle 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 , and draw a line segment connecting it to the origin. Because perpendicular lines form angles, to rotate , find a line perpendicular to the line segment connecting to the origin. Remember that perpendicular lines have opposite reciprocal slopes. will be on the perpendicular line, the same distance from the origin as was.
Once you have , you can build the rectangle around it. was at point (-3, 1) and is at point (-1, -3). In general, a counterclockwise rotation about the origin takes and maps it to .
Concept Problem Revisited
To make sure your friend correctly rotates the pentagon , you need to also tell her the center of rotation and the direction of rotation.
A rotation of about a given point takes each point on a shape and moves it to such that is on the circle with center and radius and . Point is the center of rotation.
A rigid transformation is a transformation that preserves distance and angles.
1. Describe the rotation below.
2. Rotate the rectangle 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 clockwise about the origin?
1. This is a clockwise rotation about point . You could also say this is a counterclockwise rotation about point .
2. You can see that there is a straight line passing through the origin connecting each pair of corresponding points. .
3. There would be no difference. A rotation clockwise is the same as a rotation counterclockwise.
Describe each of the following rotations in TWO ways.
4. What does a rotation look like?
5. Rotate the rectangle below clockwise about the origin.
6. Use the previous example to help you describe what happens to a point that is rotated clockwise about the origin.
7. (3, 2) is rotated clockwise about the origin. Where does it end up?
8. (3, 2) is rotated clockwise about the origin. Where does it end up? Hint: Use Guided Practice #2 for help.
9. (3, 2) is rotated counterclockwise about the origin. Where does it end up? Hint: Use Example C.
10. What do circles have to do with rotations?
11. is rotated clockwise about point to create . Describe two connections between the points , and .
For 12-15, consider below.
12. Rotate counterclockwise about point .
13. Rotate counterclockwise about point .
14. Rotate counterclockwise about point .
15. Could you have used the rule that a counterclockwise rotation takes to to perform the previous three rotations? Explain.