4.3 Vocabulary Remember…Transformation, Preimage, Image, Isometry, Congruence Transformation Rotation

A rotation is a transformation that turns a figure around a fixed point, called the center of rotation. A rotation is an isometry, so the image of a rotated figure is congruent to the preimage. This is called a congruence transformation. Positive rotations are CounterClockWise and Negative are ClockWise(+ is CCW, - is CW). Helpful Hint + -

Draw a segment from each vertex to the center of rotation Draw a segment from each vertex to the center of rotation. Your construction should show that a point’s distance to the center of rotation is equal to its image’s distance to the center of rotation. The angle formed by a point, the center of rotation, and the point’s image is the angle by which the figure was rotated.

Example 1: Drawing Rotations Copy the figure and the angle of rotation. Draw the rotation of the triangle about point Q by mA. Q A Q Step 1 Draw a segment from each vertex to point Q.

Step 3 Connect the images of the vertices. Example 1 Continued Step 2 Construct an angle congruent to A onto each segment. Measure the distance from each vertex to point Q and mark off this distance on the corresponding ray to locate the image of each vertex. Q Q Step 3 Connect the images of the vertices.

Check It Out! Example 2 Copy the figure and the angle of rotation. Draw the rotation of the segment about point Q by mX. Step 1 Draw a line from each end of the segment to point Q.

Check It Out! Example 2 Continued Step 2 Construct an angle congruent to X on each segment. Measure the distance from each segment to point P and mark off this distance on the corresponding ray to locate the image of the new segment. Step 3 Connect the image of the segment.

In this chapter all rotations will be multiples of 90̊ For a 90° rotation,+ is CCW, flip coordinates and change sign of 1st (2nd for CW rotation) OR, Just flip the coordinates and then correct the sign of each based on the quadrant it’s in

Example 3 A/B Rotate ∆RST with vertices R(–1, 4), S(2, 1), and T(3, –3) about the origin by the given angle to find R’, S’, and T'. 3A. 90° R’(–4, –1), S’(–1, 2), T’(3, 3) 3B. 180° R’(1, –4), S’(–2, –1), T’(–3, 3)

Example 4A: Drawing Rotations in the Coordinate Plane 4A Rotate ΔJKL with vertices J(2, 2), K(4, –5), and L(–1, 6) by 180° about the origin. Find J’, K’, L’: J’(-2, -2), K’(-4, -5), L’(1, -6)

Example 4B: Drawing Rotations in the Coordinate Plane 4B Rotate ∆ABC by 180° about the origin. A(2, -1), B(4, 1), C(3,3) Find A’, B’, C’: A’(-2, 1), B’(-4, -1), C’(-3, -3)

HONORS: Rotations About a Point That is Not the Origin STEPS: Transform the points to be rotated to a new coordinate system centered at the rotation point by subtracting the rotation point coordinates from the coordinates of the points to be rotated. Rotate the points in the new coordinate system with the origin at the point of rotation. Correct “signs” based on new quadrants referenced to new origin. Transform the rotated coordinates back to the original origin by adding the rotation point coordinates to the coordinates of the rotated points.

Example 5: Rotating About a Point that is Not the Origin Rotate A(6, 8), B(4, -6) 90̊ about R(3, 4) 1. Transform to R as new origin by subtracting (3, 4) from each: A’(3, 4), B’(1, -10) 2. Rotate 90̊ with R as Origin: A”(-4, 3), B”(10, 1) 3. Transform back to original system by adding (3, 4) to each: A’”(-1, 7), B’”(13, 5)