10.6: Rotating
Learning Objectives
- Find the image of a point in a rotation in a coordinate plane.
- Recognize that a rotation is an isometry.
Sample Rotations
In this lesson we will study rotations centered at the origin of a coordinate plane. We begin with some specific examples of rotations. Later we will see how these rotations fit into a general formula.
We define a rotation as follows: In a rotation centered at the origin with an angle of rotation of , a point moves counterclockwise along an arc of a circle. The central angle of the circle measures .
The original preimage point is one endpoint of the arc, and the image of the original point is the other endpoint of the arc:
- Rotations centered at the origin move points ______________________ along an arc of a circle.
- For a rotation of , the central angle of the circle measures ____________.
- The preimage point is one endpoint of the _________________ and the image is the other endpoint.
Rotation
Our first example is rotation through an angle of :
In a rotation, the image of is the point .
Notice:
- and are the endpoints of a diameter of a circle.
This means that the distance from the point to the origin (or the distance from the point to the origin) is a radius of the circle.
The distance from to the origin equals the distance from ________ to the origin.
- The rotation is the same as a “reflection in the origin.”
This means that if we use the origin as a mirror, the point is directly across from the point .
A _____________________________ is also a reflection in the origin.
In a rotation of , the coordinate and the coordinate of the __________________ become the negative versions of the values in the image.
A rotation is an isometry. The image of a segment is a congruent segment:
Use the Distance Formula:
So
- A rotation is an ___________________________, so distance is preserved.
- When a segment is rotated (or reflected in the origin), its image is a _________________________________ segment.
Rotation
The next example is a rotation through an angle of . The rotation is in the counterclockwise direction:
In a rotation, the image of is the point .
Notice:
- and are both radii of the same circle, so .
If and are both radii, then they are the same ______________________.
- is a right angle.
- The acute angle formed by and the axis and the acute angle formed by and the axis are complementary angles.
Remember, complementary angles add up to ________________.
You can see by the coordinates of the preimage and image points, in a rotation:
- the and coordinates are switched AND
- the coordinate is negative.
In a rotation, switch the _________- and _________-coordinates and make the new coordinate _________________________.
A rotation is an isometry. The image of a segment is a congruent segment.
Use the Distance Formula:
So
Reading Check:
Which of the following are isometries? Circle all that apply:
Example 1
What are the coordinates of the vertices of in a rotation of ?
Point is (4, 6), is (–4, 2), and is (6, –2).
In a rotation, the coordinate and the coordinate are switched AND the new coordinate is made negative:
- becomes : switch and to (6, 4) and make negative (–6, 4)
- becomes : switch and to (2, –4) and make negative (–2, –4)
- becomes : switch and to (–2, 6) and make negative
So the vertices of are (–6, 4), (–2, –4), and (2, 6).
Plot each of these points on the coordinate plane above and draw in each side of the new rotated triangle. Can you see how is rotated to ?
Reading Check:
1. True or false: A rotation is always in the counterclockwise direction.
2. On the coordinate plane below, create a point anywhere you like, and label it .
Then draw a second point that is the image of point rotated .
3. On the coordinate plane above, draw a third point that is the image of your original point rotated .
4. Is a rotation an isometry? Explain.
5. Is a rotation an isometry? Explain.
6. What type of rotation is the same as a reflection in the origin?
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Date Created:
Feb 23, 2012Last Modified:
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