Chapter 9Transformations Objective: Students will recognize and draw reflections, translations, dilations, and rotations.
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)
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
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
Determine how many lines of symmetry each figure has. Then decide if there is point symmetry.
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.
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)
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)
Translations Using Reflections • A translation can be found by reflecting a figure across 2 PARALLEL lines. • Each successive transformation is called a composition • Example: •
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.
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.
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
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
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
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
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
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
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)
Identify the Scale Factor, then determine if it is an enlargement, reduction, or congruence transformation. 2, enlargement ½ , reduction