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# Geom 7point2and3

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### Geom 7point2and3

1. 1. Reflections & Rotations <ul><li>Objectives: </li></ul><ul><li>Identify and use reflections </li></ul><ul><li>Identify and use rotations </li></ul>
2. 2. Reflection <ul><li>Reflection acts like a mirror. The mirror line is the line of reflection . </li></ul>
3. 3. Reflection <ul><li>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: </li></ul><ul><ul><li>If P is not on m, then m is the perpendicular bisector of PP’ </li></ul></ul><ul><ul><li>If P is on m, then P = P’ </li></ul></ul>P P’ m
4. 4. Reflection Theorem <ul><li>A reflection is an isometry. </li></ul><ul><li>What is an isometry? </li></ul><ul><li>A transformation that preserves lengths. </li></ul>
5. 5. Reflections & Symmetry <ul><li>A figure in the plane has a line of symmetry if the figure can be mapped onto itself by a reflection in the line. </li></ul><ul><li>How many lines of symmetry? </li></ul>
6. 6. Reflections & Symmetry <ul><li>A figure in the plane has a line of symmetry if the figure can be mapped onto itself by a reflection in the line. </li></ul><ul><li>How many lines of symmetry? </li></ul>
7. 7. Reflections & Symmetry <ul><li>A figure in the plane has a line of symmetry if the figure can be mapped onto itself by a reflection in the line. </li></ul><ul><li>How many lines of symmetry? </li></ul>
8. 8. Reflections & Symmetry <ul><li>A figure in the plane has a line of symmetry if the figure can be mapped onto itself by a reflection in the line. </li></ul><ul><li>How many lines of symmetry? </li></ul>
9. 9. Practice <ul><li>Do p. 407 #3-14, 41 </li></ul>
10. 10. Rotations <ul><li>A rotation is a transformation in which a figure is turned about a fixed point. </li></ul><ul><li>The fixed point is the center of rotation. </li></ul>
11. 11. Rotations <ul><li>Rays drawn from the center of rotation to a point and its image form an angle called the angle of rotation. </li></ul>
12. 12. Rotation Theorem <ul><li>A rotation is an isometry. </li></ul>
13. 13. Constructing a Rotation <ul><li>Open your books to p. 413. </li></ul><ul><li>Draw triangle ABC and point P like you see in the book. </li></ul>
14. 14. Constructing a Rotation <ul><li>1. Draw a segment connecting vertex A and the center of rotation point P. </li></ul><ul><li>2. Use a protractor to measure a 120˚ angle counterclockwise and draw a ray. </li></ul><ul><li>3. Place the point of the compass at P and draw an arc from A to locate A’. </li></ul><ul><li>Repeat steps 1-3 for each vertex. </li></ul><ul><li>Connect the vertices to form the image. </li></ul>
15. 15. Constructing a Rotation <ul><li>Plot the points: </li></ul><ul><ul><li>A: 2, -2 </li></ul></ul><ul><ul><li>B: 4, 1 </li></ul></ul><ul><ul><li>C: 5, 1 </li></ul></ul><ul><ul><li>D: 5, -1 </li></ul></ul><ul><li>Now rotate this figure 90˚ counterclockwise around the origin. </li></ul>
16. 16. Another Theorem <ul><li>Look at the picture in the middle of p. 414 </li></ul><ul><li>2 reflections = a rotation </li></ul><ul><li>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. </li></ul><ul><li>The angle of rotation is 2x˚, where x˚ is the measure of the acute or right angle formed by k and m. </li></ul>
17. 17. Look at the picture at the bottom of p. 414 <ul><li>Reflection #1: blue to red </li></ul><ul><li>Reflection #2: red to green </li></ul><ul><li>We call this a clockwise rotation of 120˚ about point P </li></ul>
18. 18. Rotational Symmetry <ul><li>If you rotate a square 90˚, what do you get? </li></ul><ul><li>If you rotate a square 180˚, what do you get? </li></ul><ul><li>This is called rotational symmetry . </li></ul><ul><li>A figure in the plane has rotational symmetry if the figure can be mapped onto itself by a rotation of 180˚ or less. </li></ul>
19. 19. Rotational Symmetry <ul><li>Does an octagon have rotational symmetry? </li></ul><ul><li>Yes, it can be mapped onto itself by a rotation in either direction of 45˚, 90˚, 135˚, or 180˚ about its center. </li></ul>
20. 20. Rotational Symmetry <ul><li>Does a parallelogram have rotational symmetry? </li></ul><ul><li>Yes, it can be mapped onto itself by a rotation of 180˚ around its center </li></ul>
21. 21. Rotational Symmetry <ul><li>Does a trapezoid have rotational symmetry? </li></ul><ul><li>No </li></ul>
22. 22. Look at Example 5 on p. 415 <ul><li>In a. (ozone), what rotational symmetry do you see? </li></ul><ul><li>What do you see in b.? </li></ul><ul><li>Do p. 7 2-12, 36-39 </li></ul>
23. 23. Homework: <ul><li>Page 407 16-28 evens </li></ul><ul><li>Page 416, 14-18 </li></ul>