12.1.3 Rotations (continued)
Example 2 Drawing Rotations
A method for drawing the counter-clockwise rotation of a preimage is demonstrated in this example. The figure, the angle of rotation, and the center of rotation are given. To draw a vertex of the image, draw a line segment from a preimage vertex to the center of rotation, P . Construct an angle congruent to the angle of rotation using the line segment as one side of the angle and with the angle vertex at P . Measure the distance from P to the preimage vertex and mark off the same distance from P along the other side, or ray, of the angle. The mark is the corresponding image vertex. Follow the same steps with each of the other vertices. Draw line segments between the image vertices to draw the rotated figure.
Example 3 Drawing Rotations in the Coordinate Plane
A method for drawing the counter-clockwise rotation of a figure in the coordinate plane is demonstrated in this example. The coordinates of the vertices and the angle of rotation are given. The center of rotation is given as the origin. The rule for this particular rotation is: ( x , y ) → ( − y , x ). This rotation changes all x distances into y distances of the opposite sign, and all y distances into x distances of the same sign. To visualize why this is the case, imagine rotating the graph 90° counterclockwise about the origin. Apply the rule to each of the vertices of the triangle. For example, A ( − 1, 1) → A ′( − 1, − 1). Draw line segments between the image vertices to draw the image figure. The rule for a 180° counterclockwise rotation about the origin is: ( x , y ) → ( − x , − y ). The rule for a 270° counterclockwise rotation about the origin is: ( x , y ) → ( y , − x ).
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