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 Translation Flip figure along the dotted line Yes No Reflection Rotate figure 90° in a counter clockwise direction 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

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

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 PointImage 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.