Transformations: Rotations: Transformations: Rotations
What is a rotation?: What is a rotation? To put it simply, a rotation is the spinning of an object around a fixed point (known as the center of rotation). The center of rotation can be the origin, a point on the shape, or a point not on the shape. Rotations can go either clockwise or counter-clockwise. The number of degrees an object rotates is called the angle of rotation.
For those who don’t know how to read an analog clock….: For those who don’t know how to read an analog clock…. Clockwise Counter-Clockwise
Rotating an Object by using a Protractor: Rotating an Object by using a Protractor Use your protractor to draw a line segment from the point of rotation to a point on the object. Use that line as the base of your angle, and then mark off the angle of rotation. Draw that line. Place your object appropriately on the new line.
Example 1: Example 1 P Rotate the shape 90 ° clockwise around the point P. Draw a line segment connecting P to two vertices. Using those line segment as a base, take your protractor and mark off 90 ° in a clockwise direction for each one, making sure that P is in the center of the protractor. (Note: the new line segments should be the same length as the original line segments.) Now, place your shape accordingly.
Example 2: Example 2 P Rotate the shape 180 ° counter-clockwise around the point P. Now, place your shape accordingly. Remember: 180 ° is a straight line!
Rotations about the origin: Rotations about the origin Using some knowledge of coordinates, we can easily draw rotations about the origin. If you rotate 90 ° clockwise around the origin, the rule is (x, y) (y, -x) If you rotate 90 ° counter-clockwise around the origin, the rule is (x, y) (-y, x) If you rotate 180 ° around the origin, the rule is (x, y) (-x, -y) To rotate around the origin, simply apply the appropriate rules to the shape’s vertices and graph.
Example 1: Example 1 Determine the coordinates of the endpoints of the line segment after the given rotation about the origin. Line segment with endpoints A(-5, 6) and B(14, 2) is rotated 90 ° clockwise around the origin. The rule is (x, y) (y, -x) A ´(6, 5) B ´(2, -14)
Example 2: Example 2 Determine the coordinates of the triangle after the given rotation about the origin. Triangle with vertices A(10, -3) and B(5, 4) and C(-2, -1) is rotated 90 ° counter-clockwise around the origin. The rule is (x, y) (-y, x) A´(3, 10) B ´(-4, 5) C´(1, 2)
Example 3: Example 3 Determine the coordinates of the triangle after the given rotation about the origin. Triangle with vertices A(2, -30) and B(13, 14) and C(60, -4) is rotated 180 ° around the origin. The rule is (x, y) (-x, -y) A´(-2, 30) B ´(-13, -14) C´(-60, 4)