Ch 9 3 rotation Draw rotated images using

Ch 9 -3

• rotation • Draw rotated images using the angle of rotation. • center of rotation • Identify figures with rotational symmetry. • angle of rotation • rotational symmetry • invariant points • direct isometry • indirect isometry

Rotations ØA transformation in which a figure is turned about a fixed point. ØThe fixed point is the Center of Rotation ØRays drawn from the center of rotation to a point and its image form an angle called the Angle of Rotation.

Watch when this rectangle is rotated by a given angle measure. Hi Center of

Angle of Rotation Hi Center of

A. For the following diagram, which description best identifies the rotation of triangle ABC around point Q? A. 20° clockwise B. 20° counterclockwise C. 90° clockwise D. 90° counterclockwise A. B. C. D. A B C D

Rotations • A composite of two reflections over two intersecting lines • The angle of rotation is twice the measure of the angle b/t the two lines of reflection • Coordinate Plane rotation Rotating about the origin

Reflections in Intersecting Lines Find the image of parallelogram WXYZ under reflections in line p and then line q. First reflect parallelogram WXYZ in line p. Then label the image W'X'Y'Z'. Next, reflect the image in line q. Then label the image W''X''Y''Z''. Answer: Parallelogram W''X''Y''Z'' is the image of parallelogram WXYZ under reflections in line p and q.

In the following diagram, which triangle is the image of ΔABC under reflections in line m and then line n. A. blue Δ B. green Δ C. neither 1. 2. 3. A B C

Coordinate Plane Rotation Rotating about the origin • Clockwise vs. Counterclockwise • 90 o Quarter turn • 180 o Half turn (clockwise or counterclockwise) • 270 o Three quarter turn Big Hint!!! If you need to rotate a shape about the origin, • TURN THE PAPER • Write down the new coordinates • Turn the paper back and graph the rotated points.

Example #1 • Rotate ABC 90 o clockwise about the origin. Turn the paper A’ (90 o clockwise) Write the new coordinates C’ B’ A’ (2, 4) B’ (4, 1) C’ (-1, 3) Turn the paper back and graph the rotated points

Example #2 • Rotate ABC 180 o about the origin. Turn the paper (180 o) Write the new coordinates A’ (4, -2) C’ B’ (1, -4) C’ (3, 1) Turn the paper back and graph the rotated points A’ B’

Rotational Symmetry • A figure has rotational symmetry if it can be mapped onto itself by a rotation of 180º or less. – Equilateral Triangle – Square – Most regular polygons A B D C

There are 3 rotations (<360 degrees) where the triangle maps onto itself. The equilateral triangle has rotational symmetry of order = 3. magnitude of symmetry An equilateral triangle maps onto itself every 120 degrees of rotation.

An regular pentagon has an order of 5. 3 2 1 2 4 5 5 4 4 3 1 5 3 1 2 2 3 3 4 1 2 4 5 5 1 1 5 2 4 magnitude of symmetry 3

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

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

Draw a Rotation B. Triangle DEF has vertices D(– 2, – 1), E(– 1, 1), and F(1, – 1). Draw the image of DEF under a rotation of 115° clockwise about the point G(– 4, – 2). First draw ΔDEF and plot point G. 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 Repeat with points E and F.

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

B. Triangle ABC has vertices A(1, – 2), B(4, – 6), and C(1, – 6). Draw the image of ΔABC under a rotation of 70° counterclockwise about the point M(– 1, – 1). A. B. C. D. A. B. C. D. A B C D