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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
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture. QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
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.
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
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