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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 origin can be described as (x, y) (-x, -y)

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Example A 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. 180° about the origin 90° clockwise about the origin 270° clockwise about the origin

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Example A triangle has vertices F(-3, 3), G(1, 3), and H(1, 1). Give the vertices of the image after the described rotations. 180° about the origin 90° clockwise about the origin 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

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Examples Work on these problems from the book… Pg 416 #1-3, 6-12

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