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Transformations 1 G.3cd Symmetry and Transformations

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Transformations 2 Lines of Symmetry If a line can be drawn through a figure so the one side of the figure is a reflection of the other side, the line is called a “line of symmetry.” Some figures have 1 or more lines of symmetry. Some have no lines of symmetry. One line of symmetry Infinite lines of symmetry Four lines of symmetry Two lines of symmetry No lines of symmetry

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Point of Symmetry Transformations 3

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Be careful!! Transformations 4

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5 Types of Transformations Reflections: These are like mirror images as seen across a line or a point. Translations ( or slides): This moves the figure to a new location with no change to the looks of the figure. Rotations: This turns the figure clockwise or counter-clockwise but doesn’t change the figure. Dilations: This reduces or enlarges the figure to a similar figure.

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Transformations 6 Reflections You could fold the picture along line l and the left figure would coincide with the corresponding parts of right figure. l You can reflect a figure using a line or a point. All measures (lines and angles) are preserved but in a mirror image. Example:The figure is reflected across line l.

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Transformations 7 Reflections – continued… reflects across the y axis to line n (2, 1) (-2, 1) & (5, 4) (-5, 4) Reflection across the x-axis: the x values stay the same and the y values change sign. (x, y) (x, -y) Reflection across the y-axis: the y values stay the same and the x values change sign. (x, y) (-x, y) Example:In this figure, line l : reflects across the x axis to line m. (2, 1) (2, -1) & (5, 4) (5, -4) ln m

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Transformations 8 Reflections across specific lines: To reflect a figure across the line y = a or x = a, mark the corresponding points equidistant from the line. i.e. If a point is 2 units above the line its corresponding image point must be 2 points below the line. (-3, 6) (-3, -4) (-6, 2) (-6, 0) (2, 3) (2, -1). Example: Reflect the fig. across the line y = 1.

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Transformations 9 Translations (slides) If a figure is simply moved to another location without change to its shape or direction, it is called a translation (or slide). If a point is moved “a” units to the right and “b” units up, then the translated point will be at (x + a, y + b). If a point is moved “a” units to the left and “b” units down, then the translated point will be at (x - a, y - b). A B Image A translates to image B by moving to the right 3 units and down 8 units. Example: A (2, 5) B (2+3, 5-8) B (5, -3)

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Transformations 10 Composite Reflections If an image is reflected over a line and then that image is reflected over a parallel line (called a composite reflection), it results in a translation. A B C Image A reflects to image B, which then reflects to image C. Image C is a translation of image A Example:

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Transformations 11 Rotations An image can be rotated about a fixed point. The blades of a fan rotate about a fixed point. An image can be rotated over two intersecting lines by using composite reflections. Image A reflects over line m to B, image B reflects over line n to C. Image C is a rotation of image A. A B C m n

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Rotate 90 counterclockwise about the origin Rotation is simply turning about a fixed point. For our purposes, the fixed point will be the origin Rotate 180 about the origin Rotate 90 clockwise about the origin

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CLOCKWISE is like a right turn.

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Hands in the air on the wheel. Left hand is x Right hand is y

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Make a clockwise turn. Which hand is at 12 o’clock first?

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Rotate 90° clockwise

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COUNTERCLOCKWISE is like a left turn.

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Hands in the air on the wheel. Left hand is x Right hand is y

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Make a counterclockwise turn. Which hand is at 12 o’clock first?

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Rotate 90° counterclockwise

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Rotate 180°

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Transformations 32 Angles of rotation In a given rotation, where A is the figure and B is the resulting figure after rotation, and X is the center of the rotation, the measure of the angle of rotation AXB is twice the measure of the angle formed by the intersecting lines of reflection. Example: Given segment AB to be rotated over lines l and m, which intersect to form a 35° angle. Find the rotation image segment KR. A B 35 °

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Transformations 33 Angles of Rotation.. Since the angle formed by the lines is 35°, the angle of rotation is 70°. 1. Draw AXK so that its measure is 70° and AX = XK. 2. Draw BXR to measure 70° and BX = XR. 3. Connect K to R to form the rotation image of segment AB. A B 35 ° X K R

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Transformations 34 Dilations A dilation is a transformation which changes the size of a figure but not its shape. This is called a similarity transformation. Since a dilation changes figures proportionately, it has a scale factor k. If the absolute value of k is greater than 1, the dilation is an enlargement. If the absolute value of k is between 0 and 1, the dilation is a reduction. If the absolute value of k is equal to 0, the dilation is congruence transformation. (No size change occurs.)

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Transformations 35 Dilations – continued… In the figure, the center is C. The distance from C to E is three times the distance from C to A. The distance from C to F is three times the distance from C to B. This shows a transformation of segment AB with center C and a scale factor of 3 to the enlarged segment EF. In this figure, the distance from C to R is ½ the distance from C to A. The distance from C to W is ½ the distance from C to B. This is a transformation of segment AB with center C and a scale factor of ½ to the reduced segment RW. C E A F B C R A B W

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Transformations 36 Dilations – examples… Find the measure of the dilation image of segment AB, 6 units long, with a scale factor of 1. S.F. = -4: the dilation image will be an enlargment since the absolute value of the scale factor is greater than 1. The image will be 24 units long. 2.S.F. = 2/3: since the scale factor is between 0 and 1, the image will be a reduction. The image will be 2/3 times 6 or 4 units long. 3.S.F. = 1: since the scale factor is 1, this will be a congruence transformation. The image will be the same length as the original segment, 1 unit long.

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