Reflections & Rotations

Objectives:Identify and use reflectionsIdentify and use rotations

Reflection

Reflection acts like a mirror. The mirror line is the line of reflection.

Reflection

A reflection in a line m is a transformation that maps every point P in the plane to a point P’ so that the following properties are true: If P is not on m, then m is the perpendicular

bisector of PP’ If P is on m, then P = P’ P

P’

m

Reflection Theorem

A reflection is an isometry.What is an isometry? A transformation that preserves lengths.

Reflections & Symmetry

A figure in the plane has a line of symmetry if the figure can be mapped onto itself by a reflection in the line.

How many lines of symmetry?

Reflections & Symmetry

A figure in the plane has a line of symmetry if the figure can be mapped onto itself by a reflection in the line.

How many lines of symmetry?

Reflections & Symmetry

A figure in the plane has a line of symmetry if the figure can be mapped onto itself by a reflection in the line.

How many lines of symmetry?

Reflections & Symmetry

A figure in the plane has a line of symmetry if the figure can be mapped onto itself by a reflection in the line.

How many lines of symmetry?

Practice

Do p. 407 #3-14, 41

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Rotations

A rotation is a transformation in which a figure is turned about a fixed point.

The fixed point is the center of rotation.

Rotations

Rays drawn from the center of rotation to a point and its image form an angle called the angle of rotation.

Rotation Theorem

A rotation is an isometry.

Constructing a Rotation

Open your books to p. 413.Draw triangle ABC and point P like you

see in the book.

Constructing a Rotation

1. Draw a segment connecting vertex A and the center of rotation point P.

2. Use a protractor to measure a 120˚ angle counterclockwise and draw a ray.

3. Place the point of the compass at P and draw an arc from A to locate A’.

Repeat steps 1-3 for each vertex.Connect the vertices to form the image.

Constructing a Rotation

Plot the points: A: 2, -2 B: 4, 1 C: 5, 1 D: 5, -1

Now rotate this figure 90˚ counterclockwise around the origin.

Another Theorem

Look at the picture in the middle of p. 4142 reflections = a rotationIf lines k and m intersect at point P, then a

reflection in k followed by a reflection in m is a rotation about point P.

The angle of rotation is 2x˚, where x˚ is the measure of the acute or right angle formed by k and m.

Look at the picture at the bottom of p. 414

Reflection #1: blue to redReflection #2: red to greenWe call this a clockwise rotation of 120˚

about point P

Rotational Symmetry

If you rotate a square 90˚, what do you get?

If you rotate a square 180˚, what do you get?

This is called rotational symmetry.A figure in the plane has rotational

symmetry if the figure can be mapped onto itself by a rotation of 180˚ or less.

Rotational Symmetry

Does an octagon have rotational symmetry?

Yes, it can be mapped onto itself by a rotation in either direction of 45˚, 90˚, 135˚, or 180˚ about its center.

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Rotational Symmetry

Does a parallelogram have rotational symmetry?

Yes, it can be mapped onto itself by a rotation of 180˚ around its center

Rotational Symmetry

Does a trapezoid have rotational symmetry?

No

Look at Example 5 on p. 415

In a. (ozone), what rotational symmetry do you see?

What do you see in b.?Do p. 7 2-12, 36-39

Homework:

Page 407 16-28 evens Page 416, 14-18