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7.3 Rotations
Advanced Geometry
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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.
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Angle of Rotation
Rays drawn from the center of rotation to a point and its image form an angle called the angle of rotation.
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Let’s Draw a Rotation
First draw a triangle ABC and a point outside the triangle called P.
We are going to draw a rotation of 120 degrees counterclockwise about P.
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Next…
Draw a segment connecting vertex A and the center of rotation point P.
We will need to know the measurement of this segment later…might want to find that now.
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Next….
Use a protractor to measure a 120 degree angle counterclockwise and draw a ray.
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Next….
We want to cut that ray down so that it is congruent to segment PA. Name the endpoint A’.
You could also do this with a compass
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Next….
Repeat the steps for points B and C.
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Example
A quadrilateral has vertices P(3,-1), Q(4,0), R(4,3), and S(2,4). Rotate PQRS 180 degrees counterclockwise about the origin and name the coordinates of the new vertices.
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Example
Triangle RST has coordinates R(-2,3), S(0,4), and T(3,1). If triangle RST is rotated 90 degrees clockwise about the origin. What are the coordinates of the new vertices?
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Rotation Tricks
Rotation of 90° clockwise or ___ ° counterclockwise about the origin can be
described as
(x, y) (y, -x)
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Rotation Tricks
Rotation of 270° clockwise or ___ ° counterclockwise about the origin can be
described as
(x, y) (-y, x)
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Rotation Tricks
Rotation of 180° about the origincan be described as
(x, y) (-x, -y)
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ExampleA quadrilateral has vertices A(-2, 0), B(-3, 2), C(-2, 4), and D(-1, 2). Give the vertices of the image after the described rotations.
1. 180° about the origin
2. 90° clockwise about the origin
3. 270° clockwise about the origin
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ExampleA triangle has vertices F(-3, 3), G(1, 3), and H(1, 1). Give the vertices of the image after the described rotations.
1. 180° about the origin
2. 90° clockwise about the origin
3. 270° clockwise about the origin
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Theorem 7.3
If 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.
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Example
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Example
If m and n intersect at Q to form a 30° angle. If a pentagon is reflected in m, then in n about point Q, what is the angle of rotation of the pentagon?
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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.
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Rotational Symmetry
Which figures have rotational symmetry? For those that do, describe the rotations that map the figure onto itself.
1. 2.
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Examples
Work on these problems from the book…
Pg 416 #1-3, 6-12