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Introduction to Transformations in Math

Learn about Reflections, Translations, Rotations, and Dilations in geometry. Understand how figures change across different lines or points through various transformations.

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Introduction to Transformations in Math

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  1. Lesson 10-5 Transformations Lesson 10-5: Transformations

  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. Lesson 10-5: Transformations

  3. Reflections You can reflect a figure using a line or a point. All measures (lines and angles) are preserved but in a mirror image. You could fold the picture along line l and the left figure would coincide with the corresponding parts of right figure. Example: The figure is reflected across line l . l Lesson 10-5: Transformations

  4. Reflections – continued… Reflection across the x-axis: the x values stay the same and the y values change sign. (x , y)  (x, -y) • reflects across the y axis to line n (2, 1)  (-2, 1) & (5, 4)  (-5, 4) 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 : n l • reflects across the x axis to line m. (2, 1)  (2, -1) & (5, 4)  (5, -4) m Lesson 10-5: Transformations

  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. Example: Reflect the fig. across the line y = 1. (2, 3)  (2, -1). (-3, 6)  (-3, -4) (-6, 2)  (-6, 0) Lesson 10-5: Transformations

  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. Four lines of symmetry One line of symmetry Two lines of symmetry Infinite lines of symmetry No lines of symmetry Lesson 10-5: Transformations

  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). Example: A Image A translates to image B by moving to the right 3 units and down 8 units. B A (2, 5) B (2+3, 5-8)  B (5, -3) Lesson 10-5: Transformations

  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. Example: C B A Image A reflects to image B, which then reflects to image C. Image C is a translation of image A Lesson 10-5: Transformations

  9. A C m B n 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. Lesson 10-5: Transformations

  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) Ex: (1,2) (-2,1) & (6,2)  (-2, 6) 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) Lesson 10-5: Transformations

  11. B A 35 ° 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. Lesson 10-5: Transformations

  12. K B A R X 35 ° 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. Lesson 10-5: Transformations

  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.) Lesson 10-5: Transformations

  14. E A F C B A R B C W 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. Lesson 10-5: Transformations

  15. Dilations – examples… Find the measure of the dilation image of segment AB, 6 units long, with a scale factor of • 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. • 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. • 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. Lesson 10-5: Transformations

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