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TRANSFORMATIONS – ROTATION EXPLORATION

Author:Suzanne Harper

Part 1: Rotating Quadrilaterals

Observe the quadrilaterals in the sketch below. The red quadrilateral has been rotated about the marked center point; its image is gray.

1. Identify the angle of rotation used to transform the red figure to the gray figure.  Describe your method for determining the angle. 

2. How can we be sure that the angle measure you identified is correct?

3. Drag the vertices and sides of the red figure, observe and record how the image of the figure changes.  Drag the center point and describe how the figures move.

Part 2: Rotating Triangles

1. In the sketch below, construct a point P, and ΔABC so that point P lies on the exterior of ΔABC. 
2. If you were to rotate ΔABC by 180° about center point P, where would the triangle lie? Be explicit, and place prediction points in your sketch window.
3. Rotate ΔABC (both its sides and vertices) by 180° about center point P. [Technology Tip: Along the tool bar, under the third button from the right, select Rotate about Point; select the triangle ABC; then the center of rotation, and in the dialog box, enter 180 and click OK.]

4. Drag the vertices and sides of the original figure and observe how the triangles are related, also drag the center point P.

5. Move the center point so that it is located in the interior of the triangle. Describe the location of ΔA'B'C'. How does the location of P relate to ΔA'B'C'?

Part 3: Properties of Rotations

1. In the sketch below, construct segments connecting vertex A of the original triangle and its image to the center point: A to P, and A' to P.  Make these segments red. [Technology Tip: Select a segment; and select Set color button under the top-right tool bar.]

2. Drag the vertices and sides of the triangle around and describe the relationship between the angle formed by the red segments and the rotation.

3. Construct segments connecting vertex B of the original triangle and its image to the center point: B to P, and B' to P.  Make these segments blue. 4. Construct segments connecting vertex C of the original triangle and its image to the center point: C to P, and C' to P.  Make these segments green.

5. How is the rotation related to an angle formed by segments connecting avertex and its rotated image?

Part 4: Locating the Center of Rotation

1. Observe the two quadrilaterals in the sketch below. Predict the center and angle of rotation that transforms the gray quadrilateral to the blue quadrilateral. Place a point where you think the center of the rotation is in the sketch window.

2. Construct both the center and angle of rotation that transforms the gray quadrilateral to the blue quadrilateral. By any means possible. Describe your process below.

3. Predict the center and angle of rotation that transforms the pink triangle to the blue triangle. Place a point where you think the center of the rotation is in the sketch window.

4. Construct both the center and angle of rotation that transforms the pink triangle to the blue triangle without moving any of the figures in your sketch window. Describe your method of finding the angle and center of rotation.

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