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Chapter 9 Transformations

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  1. Chapter 9Transformations Objective: Students will recognize and draw reflections, translations, dilations, and rotations.

  2. Transformations • A transformation maps an initial figure, called the preimage, onto a final figure, called the image. • Four main transformations • Reflection • Translation • Dilation • Rotation • Isometry: a congruence transformation (nothing changes but the location)

  3. Reflections • A reflection is a transformation representing a flip of a figure. • A line of reflection is the line that a figure is “flipped” over. • Each point will be the same distance away from the line of reflection

  4. Reflect the figure over a line.

  5. Reflections in the Coordinate Plane

  6. Examples

  7. Examples

  8. Lines and Points of Symmetry • When a figure can be folded so that the 2 halves match exactly, the fold like is a line of reflection called the line of symmetry • For some figures, a point can be found that is a common point of reflection for all points on a figure. This is the point of symmetry

  9. Determine how many lines of symmetry each figure has. Then decide if there is point symmetry.

  10. Translations • A translation is a transformation that moves ALL points of a figure the same distance and direction. • Translations represent a slide of a figure.

  11. Translations in the coordinate plane • To translate points in the coordinate plane, a specific translation will be given. • (x,y)  (x+a,y+b) where a and b are fixed #s • Example: • A(3,2) through (x,y)  (x+1, y-3) = A’(4,-1)

  12. Write the translation for each transformation, then find each new point under the translation. (x,y)  (x-2, y+1) ; P’(-3,4), Q’(0,3) (x,y)  (x+3, y+1) ; S’(3,3), Q’(6,2), U’(5,-1), R’(2,0)

  13. Translations Using Reflections • A translation can be found by reflecting a figure across 2 PARALLEL lines. • Each successive transformation is called a composition • Example: • 

  14. Rotations • A rotation is a transformation that turns every point of the preimage through a specified angle and direction. • The fixed point is the center of rotation. • The degree of the turn is the angle of rotation.

  15. Rotations Using Reflections • A rotation can be performed by reflecting a figure in 2 INTERSECTING lines • The angle of rotation is twice the measure of the acute or right angle formed by the intersecting lines. • Reflecting an image in 2 perpendicular lines creates a 180 degree rotation.

  16. Rotations in the Coordinate Plane • Rotations in the coordinate plane will be done in either 90, 180 or 270 degrees, clockwise or counterclockwise. • Notice: • 900 ccw = 2700 cw • 2700 cw = 900 ccw • 1800 ccw = 1800 cw

  17. Rotational Symmetry • If a figure can be rotated less than 360 degrees about a point so that the image is indistinguishable from the preimage, there is rotation symmetry. • ORDER: the number of rotations less than 360 degrees • MAGNITUDE: 360 divided by the order

  18. Examples • A regular polygon always has rotational symmetry. • Find the order for a hexagon. • What is the magnitude? • A ferris wheel’s motion is an example of rotation. A certain Ferris wheel has 20 cars. • Identify the order and magnitude • What is the angle of rotation when seat 1 moves to seat 5? • If seat 1 moves 144 degrees, what seat does it now occupy? 6 60 degrees Order = 20, mag = 18 deg. 72 degrees Seat 9

  19. Dilations • A dilation is a transformation that may change the size of a figure. • Dilations are similarity transformations • If |r| > 1, enlargement • If 0 < |r| < 1, reduction • If |r| = 1, congruence transformation

  20. Dilations • To perform a dilation, multiply the measure of the preimage by the scale factor. • To find a scale factor, divide the measure of the image (new) by the preimage (old). • If the scale factor is a negative, it just flips the figure across the fixed point of dilation

  21. Find the measure of the image or preimage using the given scale factor.(do NOT multiply or divide by the negative!) • CD = 15, r = 3 • C’D’ = 7, r = - ¼ • C’D’ = 8, r = ¾ • CD = 16, r = - ½ C’D’ = 45 CD = 28 CD = 32/3 C’D’ = 8

  22. Dilations in the Coordinate Plane-DO multiply by the negative scale factor! • If P(2,4) and a scale factor of 2, P’=? • P’(4,8) • If Q(-8, 7) and a scale factor of -1.5, Q’ =? • Q’(12, -10.5)

  23. Identify the Scale Factor, then determine if it is an enlargement, reduction, or congruence transformation. 2, enlargement ½ , reduction