ROTATIONS - Mr. Arwe's Pre-AP Geometry



ROTATIONS

A rotation is the spinning of a figure or point around one central point. Rotations are described by the amount of degrees the figure is spun. To figure that amount, measure the angle created by an original point, the center of the rotation, and the image point. The measure of that angle is the degree of the rotation.

In the coordinate plane we will only consider rotations of 90 degrees, 180 degrees, and 270 degrees. These will take our original figure from quadrant I into quadrants II, III and IV, respectively. We will assume that our rotations spin counterclockwise, always using the origin as the central point. This point never changes location—it is called a fixed point.

Example 1 – The center of rotation is the origin.

A(__,__) → A’ (__, __)

B(__,__) → B’(__, __)

C(__, __) →C’(__, __)

OC = _____; OC’= _____

m[pic]COC’ = _____

The degree of this rotation is ____.

Find the slope of each segment.

[pic]= ____ [pic] = ____

[pic] = ____ [pic] = ____

[pic] = ____ [pic] = ____

Image segment is __________

to pre-image segment.

Example 2 – The center of rotation is the origin.

A(__,__) → A’ (__, __)

B(__,__) → B’(__, __)

C(__, __) →C’(__, __)

OC = _____; OC’= _____

m[pic]COC’ = _____

The degree of this rotation is ____.

Find the slope of each segment.

[pic]= ____ [pic] = ____

[pic] = ____ [pic] = ____

[pic] = ____ [pic] = ____

Image segment is __________

to pre-image segment.

Example 3 – The center of rotation is the origin.

A(__,__) → A’ (__, __)

B(__,__) → B’(__, __)

C(__, __) →C’(__, __)

OC = _____; OC’= _____

m[pic]COC’ = _____

The degree of this rotation is ____.

Find the slope of each segment.

[pic]= ____ [pic] = ____

[pic] = ____ [pic] = ____

[pic] = ____ [pic] = ____

Image segment is __________

to pre-image segment.

Functional notation Equations

A 90( rotation is described by: f(x,y) = (_____, _____) or x( = _____ y( = _____

A 180( rotation is described by: f(x,y) = (_____, _____) or x( = _____ y( = _____

A 270( rotation is described by: f(x,y) = (_____, _____) or x( = _____ y( = _____

Under a rotation:

1. Size is preserved - every segment is mapped into a segment congruent to the original segment.

2. Shape is preserved - every angle is mapped into an angle congruent to the original angle.

3. Image figure is congruent to pre-image figure—a.k.a. “Isometry”

In-Class Exercises

A transformation is described by f(x,y) = (-y,x).

1. Is this a rotation of 90, 180, or 270 degrees?

2. Under this rotation, the image of (5,2) is _________.

3. The image of (-4,3) is _________.

4. The preimage of (6,-2) is ___________.

5. If A(3,-6) ( A((-6,-3), what is the degree of rotation? __________

6. If B(-1,-4) is rotated 180(, then what are the coordinates of the image B(? _______

Geometry Name_____________________________________

Worksheet 13.7-Rotations Date______________________Period__________

1. The transformation f(x,y) = (y,-x) is what degree rotation?

Use the rotation described by f(x,y) = (y,-x). Find the image of the following points:

|(4,5) |(1,0) |(2,3) |(6,5) |

2. The transformation f(x,y) = (-y,x) is what degree rotation?

3. The transformation f(x,y) = (-x,-y) is what degree rotation?

4. Are there any fixed points under a rotation? If so, what are they?

Describe in functional notation the rotation that maps the first point onto its image. What degree rotation are they?

|(7,4) ( (-7,-4) |(3,2) ( (-2,3) |(0,2) ( (0,-2) |(5,6) ( (6,-5) |

Given the image point P(-2,5). What would the preimage be before a rotation of:

|180( |90( |270( |

|Graph the triangle created by points T(2,1); I(6,1), and E(4,4). Graph its|Graph the parallelogram created by points D(1,1); E(4,1); S(3,5); and |

|image after rotating it 90 degrees; 180 degrees; 270 degrees. |K(5,5). Graph its image after rotating it 90 degrees; 180 degrees; 270 |

| |degrees. |

18. Given points A(2,1) and B(6,9) find the following:

| |Original Points |90( rotation |180( rotation |270( rotation |

| |A(_____,_____) |A((_____,_____) |A((_____,_____) |A((_____,_____) |

| |B(_____,_____) |B((_____,_____) |B((_____,_____) |B((_____,_____) |

|Find the slope of the | | | | |

|segment | | | | |

|How does the slope compare| | | | |

|to slope of the original | | | | |

|segment? |Not applicable | | | |

|Find the equation of the | | | | |

|line containing the | | | | |

|segment | | | | |

|Are any of the equations the same? If so, which ones? |

| |

|Are any of the lines parallel? If so, which ones? |

| |

|Are any of the lines perpendicular? If so, which ones? |

| |

|Find the length of each | | | | |

|segment | | | | |

|Find the midpoint of each | | | | |

|segment | | | | |

-----------------------

[pic]

[pic]

[pic]

Ï%A

Ï%C

B Ï%

BÏ%

CÏ%

Ï%A

Ï%

O

m[pic]ABC = _____ m[pic]A B C = ____

AB = ____ A B = _____

BC = ____ B C = _____

Are size and shape preserved●A’

●C’

B’●

B●

C●

●A

●

O

m[pic]ABC = _____ m[pic]A’B’C’= ____

AB = ____ A’B’ = _____

BC = ____ B’C’ = _____

Are size and shape preserved? _____

[pic]

m[pic]ABC = _____ m[pic]A’B’C’= ____

AB = ____ A’B’ = _____

BC = ____ B’C’ = _____

Are size and shape preserved? _____

[pic]

m[pic]ABC = _____ m[pic]A’B’C’= ____

AB = ____ A’B’ = _____

BC = ____ B’C’ = _____

Are size and shape preserved? _____

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