1.6 Rotations and Rotational Symmetry

Use the points G(2, -4) and H(-6, -6) to answer the following:1. Find the slope of 2. Find the midpoint of

3. Find GH

Warm Up

Objectives

Identify and draw rotations.

Identify and describe symmetry in geometric figures.

1.6 Rotations and Rotational Symmetry

When a figure is rotated about a point, every point on the original figure has a corresponding point on the rotated image. A point and its corresponding point are the same distance from the center of rotation. The angles formed by connecting each point and its corresponding point to the center of the rotation are all congruent. The rotated figure is congruent to the original figure and has the same orientation.

No; the figure appearsto be flipped.

Yes; the figure appearsto be turned around a point.

A. B. C.

Yes, the figure appears to be turned around a point.

Example 1

How to graph a rotation: Graph trapezoid ABCD with vertices A(1, 3), B(4, 4), C(4, 0), and D(1, 1). Then graph the image of trapezoid ABCD after a rotation 90 counterclockwise about the origin and write the coordinates of its vertices.

Step 1: Graph the given points to make the given shape.

Step 2: Turn your paper the indicated rotation, and write down the new coordinates of the point when the paper is turned.y

xo

A

B

CD

A’ (-3, 1)B’ (-4, 4)C’ (0, 4)D’ (-1, 1)

A’o

x

y

D’

B’ C’

A

B

C

D

Step 3: Turn your paper back the right way and plot and label the new points that you wrote down. (You may have to turn your head to read sideways or upside down )

Example 2Graph the figure with the given vertices. Then graph the image of the figure after the indicated rotation about the origin and write the coordinates of its vertices. 1. Triangle GHI with vertices G(1, 0) 2. Polygon TUVW with vertices T(2, -4),H(3, 1), and I(2, 5); 90 counterclockwise U(3, -1), V(-1, 0), and W(-2, -3); 180

Example 3Rotate ΔJKL with vertices J(2, 2), K(4, –5), and L(–1, 6) by 180° about the origin.

Example 4Rotate ∆ABC with vertices A(2, -1), B(4, 1), and C(3, 3) by 90° about the origin.

The angle of rotational symmetry is the smallest angle through which a figure can be rotated to coincide with itself. The number of times the figure coincides with itself as it rotates through 360° is called the order of the rotational symmetry.

Angle of rotational symmetry: 90° Order: 4

Example 5Tell whether each figure has rotational symmetry. If so, give the angle of rotational symmetry and the order of the symmetry.

no rotational symmetry

yes; 90°;order: 4

A. B. C.

yes; 180°; order: 2

yes; 120°; order: 3 yes; 180°; order: 2 no rotational symmetry

a. b. c.

Example 6Tell whether each figure has rotational symmetry. If so, give the angle of rotational symmetry and the order of the symmetry.

Example 7Describe the symmetry of each icon. Copy each shape and draw any lines of symmetry. If there is rotational symmetry, give the angle and order.

No line symmetry; rotational symmetry; angle of rotational symmetry: 180°; order: 2

Line symmetry and rotational symmetry; angle of rotational symmetry: 90°; order: 4

line symmetry and rotational symmetry; 51.4°; order: 7