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S ECTION 9.3 Rotations

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In Lesson 4.7, you learned that a rotation or turn moves every point of a preimage through a specified angle and direction about a fixed point. The direction of rotation can be either clockwise or counterclockwise. Assume that all rotations are counterclockwise unless stated otherwise.

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Draw a segment from point R to point A. Locate point R' so that AR = AR'. Example 1: Rotate quadrilateral RSTV 45° counterclockwise about point A. Repeat this process for points S, T, and V. Connect the four points to form R'S'T'V'. Use a protractor to measure a 45° angle counterclockwise with as one side. Extend the other side to be longer than AR.

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Quadrilateral R'S'T'V' is the image of quadrilateral RSTV under a 45° counterclockwise rotation about point A.

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When a point is rotated 90 , 180 , or 270 counterclockwise about the origin, you can use the following rules:

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Example 2: Triangle DEF has vertices D(–2, –1), E(–1, 1), and F(1, –1). Graph ΔDEF and its image after a rotation of 115° clockwise about the point G(–4, –2). First, draw ΔDEF and plot point G. Repeat with points E and F. Draw a segment from point G to point D. Use a protractor to measure a 115° angle clockwise with as one side. Draw Use a compass to copy onto Name the segment

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ΔD'E'F' is the image of ΔDEF under a 115° clockwise rotation about point G.

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Example 3: Hexagon DGJTSR is shown below. What is the image of point T after a 90 counterclockwise rotation about the origin? Multiple Choice: a) (5, –3) b) (–5, –3) c) (–3, 5) d) (3, –5) Explanation on next slide!

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Read the Test Item You are given a graph of hexagon DGJTSR and asked to identify the coordinates of the image of point T after a 90° counterclockwise rotation about the origin. Solve the Test Item To find the coordinates of point T after a 90 counterclockwise rotation about the origin, multiply the y-coordinate by –1 and then interchange the x- and y-coordinates. (x, y) → (–y, x) (5, 3) → (–3, 5) Answer: The answer is C, (–3, 5).

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Example 4: Triangle PQR is shown below. What is the image of point Q after a 90° counterclockwise rotation about the origin? To find the coordinates of point Q after a 90 counterclockwise rotation about the origin, multiply the y-coordinate by –1 and then interchange the x- and y-coordinates. (x, y) → (–y, x) (4, 5) → (–5, 4)

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