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You will learn to:

Draw rotated images using the angle of rotation.

Identify figures with rotational symmetry.

9-3 Objectives

Rotation

Center of Rotation

Angle Rotation

Rotational Symmetry

Vocabulary

Remember that a rotation is a transformation that turns every point of a preimage through a specified angle and direction 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.

Rotation

This is a flip, not a rotation.

This is a 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. This is called the angle of rotation.

Angle of Rotation

Q

Center of

rotation

Angle of

rotation

Relation

Rotations will be either:Counterclockwise or Clockwise.

Helpful Hint

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.

Step 3 Connect the images of the vertices.

Q

Q

Step 1 Draw a segment from each

vertex to point Q.

Q

Draw the rotation of the triangle about point Q by m A.Q

A

Drawing a Rotation

Coordinate Plane Rotations

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 DEF and plot point G.

Use a protractor to measure a 115°angle clockwise with as one side.

Use a compass to copy onto Name the segment

Draw

Repeat with points E and F.

Draw a segment from point G to point D.

D

E

FG

R

115

D'

F'

E'

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

Connect the points.

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

Answer:

Key Concepts

Postulate 9.1 In a given rotation, if A is the preimage, A’’ is

image, and P is the center of rotation, then the

measure of the angle of rotation APA’’ is twice

the measure of the acute or right angle formed by

the intersecting lines of reflection.

Corollary 9.1 Reflecting an image successively in two

perpendicular lines results in a 180 rotation.

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''.

Parallelogram W''X''Y''Z'' is the image of parallelogram WXYZ under reflections in line p and q.

Answer:

Find the image of ABC under reflections in line m and then line n.

Answer:

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

Rotational Symmetry

QUILTS Use the quilt by Judy Mathieson shown below. Identify the order and magnitude of the symmetry in the medium star directly to the left of the large star in the center of the quilt.

Answer: The medium star directly to the left of the large star in the center has rotational symmetry of order 16 and a magnitude of 22.5°.

QUILTS Use the quilt by Judy Mathieson shown below. Identify the order and magnitude of the symmetry in the tiny star above the medium-sized star in Example 3a.

Answer: The tiny star has rotational symmetry of order 8 and magnitude of 45°.

QUILTS Use the quilt by Judy Mathieson shown below. Identify the order and magnitude of the symmetry in each part of the quilt.

a. star in the upper left corner

Answer: 8; 45°

b. medium-sized star directly in center of quilt

Answer: 20; 18°

What did you learn today?

How to:

Draw rotated images using the angle of rotation.

Identify figures with rotational symmetry.

Assignment:

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