Transformations lower secondary fil..ppt
- 2. 2 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.
- 3. 3 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 .
- 4. 4 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) l n m
- 5. 5 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.
- 6. 6 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
- 7. 7 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)
- 8. 8 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:
- 9. 9 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
- 10. 10 Rotations It is a type of transformation where the object is rotated around a fixed point called the point of rotation. When a figure is rotated 90° counterclockwise about the origin, switch each coordinate and multiply the first coordinate by -1. (x, y) (-y, x) When a figure is rotated 180° about the origin, multiply both coordinates by -1. (x, y) (-x, -y) Ex: (1,2) (-1,-2) & (6,2) (-6, -2) Ex: (1,2) (-2,1) & (6,2) (-2, 6)
- 11. 11 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 °
- 12. 12 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
- 13. 13 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.)
- 14. 14 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
- 15. 15 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.
- 16. KutaSoftware: Geometry- All Transformations Part 1 https://www.youtube.com/watch?v=w33 N9ZWj4Tk 16
- 17. KutaSoftware: Geometry- All Transformations Part 2 https://www.youtube.com/watch?v=guav zJ7tXj0 17