rotations. goals distinguish between a translation, reflection, and rotation. visualize, and then...
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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.
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