Rotations

Goals

• Distinguish between a translation, reflection, and rotation.

• Visualize, and then perform rotations using patty paper.

• To determine the coordinates of a rotated point on a coordinate grid centered at the origin.

Visualizing Rotations

Slide figure to right a length equal to the bottom of the triangle

Yes

Yes

Translation

Flip figure along the dotted line

Yes

No

Reflection

Rotate figure 90° in a counter clockwise direction

Yes

Yes

Rotation

Visualizing Rotations

Rotations

• Draw this shape on patty paper with pencil markings on both sides of the paper.

• For each figure and point of rotation C, visualize where the image will be.

• Perform the transformation using patty paper and record.

• Shade the original figure blue and its image red.

Rotations

Rotations

Properties of Rotations• Rotations preserve distance.• Rotations preserve parallelism.• Rotations preserve angle measure.• Rotations maintain orientation.• Rotations preserve collinearity.• Rotations preserve betweenness.• A rotation is a transformation such that the image of every point

is a specified angle from a fixed point, called the center point of the rotation.

• A rotation is a transformation such that the image of every point is a specified distance from a fixed point, called the center point of the rotation.

Practice with Rotations

Practice with Rotations

Investigating Rotations Using Coordinates

• Open GeoGebra• Construct a polygon• Rotate the polygon

Investigating Rotations Using Coordinates

• Create a slider for the angle of rotation• Construct a polygon• Rotate the polygon

Point of rotation, vertex C

Angle of rotation

Point of rotation

Angle of rotation

Point of rotation, midpoint F

Angle of rotation

Investigating Rotations Using Coordinates

• Open a new file in GeoGebra• Create an angle slider

Investigating Rotations Using Coordinates

• Put a point at the origin (this will be the point of rotation).

• Construct a triangle with vertices – (1,2)– (3,6)– (1,7)

• Rotate the polygon about the origin by the slider

Investigating Rotations Using Coordinates

• Change the angle slider to 90°• Connect the corresponding vertices with

segments.• What are the coordinates of the vertices of

the image?Coordinates of Original Figure

Explanation of Transformation

Coordinates of Image

Observations

(1,2)

(3,6)

(1,7)

90° Counter-clockwise rotation about the origin

(-2,1)

(-6,3)

(-7,1)

• Opposite of y-coordinate becomes x-coordinate

• x-coordinate becomes y-coordinate

Investigating Rotations Using Coordinates

• Change the angle slider to 180°• What are the coordinates of the vertices of

the image?

Coordinates of Original Figure

Explanation of Transformation

Coordinates of Image

Observations

(1,2)

(3,6)

(1,7)

180° counter-clockwise rotation about the origin

(-1,-2)

(-3,-6)

(-1,-7)

• Each coordinate becomes it’s opposite

• The segments between the corresponding vertices are not parallel

Investigating Rotations Using Coordinates

• Change the angle slider to 270°• What are the coordinates of the vertices of

the image?

Coordinates of Original Figure

Explanation of Transformation

Coordinates of Image

Observations

(1,2)

(3,6)

(1,7)

270° counter-clockwise rotation about the origin

(2,-1)

(6,-3)

(7,-1)

• Opposite of x-coordinate becomes y-coordinate

• y-coordinate becomes x-coordinate

How else can I describe this rotation?

Rotations in the Coordinate Plane

• Algebraically, what is the image of the point (x,y) if it is reflected over:

Degree of Rotation (Counter-clockwise)

Original Point Image

90° (x,y)180° (x,y)270° (x,y)360° (x,y)

(-y,x)

(-x,-y)

(y,-x)

(x,y)

Properties of Rotations

Are the following preserved?– Distance– Parallelism– angle measure– Orientation– betweenness of points– collinearlity

✔

✔

✔

✖

✔

✔

Goals

• Distinguish between a translation, reflection, and rotation.

• Visualize, and then perform rotations using patty paper.

• To determine the coordinates of a rotated point on a coordinate grid centered at the origin.