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Lesson 9-1 Reflections or Flips Objectives • Draw reflected images • • • • Across x-axis Across y-axis Across origin Across line y = x • Recognize and draw lines of symmetry and points of symmetry • How many lines of symmetry in a figure? • How many points of symmetry in a figure? Vocabulary • Reflection – is a transformation representing a flip of a figure; figure may be reflected in a point, a line, or a plane. • Isometry – a congruence transformation (distance, angle measurement, etc preserved) • Line of symmetry – line of reflection that the figure can be folded so that the two halves match exactly • Point of symmetry – midpoint of all segments between the pre-image and the image; figure must have more than one line of symmetry Reflections Across the x-axis y B’ A’ Multiply y coordinate by -1 Across the origin A B x B C’ x C’ A B’ B y y A’ A C C Across the y-axis x C KEY: Equal Distance from Reflection Line Multiply x coordinate by -1 Across the line y = x y B A B’ C C’ A’ x C’ B’ A’ Multiply both coordinates by -1 Interchange x and y coordinates Common reflections in the coordinate plane Reflection x-axis y-axis origin y=x Pre-image to (a, b) (a, -b) (a, b) (-a, b) (a, b) (-a, -b) (a, b) (b, a) image Find Multiply y Multiply x Multiply both Interchange coordinates coordinate coordinate coordinates by -1 x and y by -1 by -1 coordinates A line of symmetry is like a line of reflection. The line of symmetry in a figure is a line where the figure could be folded in half so that the two halves match exactly Draw the reflected image of quadrilateral WXYZ in line p. Step 1 Draw segments perpendicular to line p from each point W, X, Y, and Z. Step 2 Locate W', X', Y', and Z' so that line p is the perpendicular bisector of Points W', X', Y', and Z' are the respective images of W, X, Y, and Z. Step 3 Connect vertices W', X', Y', and Z'. Answer: Since points W', X', Y', and Z' are the images of points W, X, Y, and Z under reflection in line p, then quadrilateral W'X'Y'Z' is the reflection of quadrilateral WXYZ in line p. Draw the reflected image of quadrilateral ABCD in line n. Answer: COORDINATE GEOMETRY Quadrilateral ABCD has vertices A(1, 1), B(3, 2), C(4, –1), and D(2, –3). Graph ABCD and its image under reflection in the x-axis. Compare the coordinates of each vertex with the coordinates of its image. Use the vertical grid lines to find the corresponding point for each vertex so that the x-axis is equidistant from each vertex and its image. D' A(1, 1) A' (1, –1) C' B(3, 2) B' (3, –2) C(4, –1) C' (4, 1) D(2, –3) D' (2, 3) A' B' Answer: The x-coordinates stay the same, but the y-coordinates are opposite. That is, (a, b) (a, –b). COORDINATE GEOMETRY Quadrilateral ABCD has vertices A(1, 1), B(3, 2), C(4, –1), and D(2, –3). Graph ABCD and its image under reflection in the y-axis. Compare the coordinates of each vertex with the coordinates of its image. Use the horizontal grid lines to find the corresponding point for each vertex so that the y-axis is equidistant from each vertex and its image. B' A(1, 1) A' (–1, 1) A' B(3, 2) B' (–3, 2) C(4, –1) C' (–4, –1) D(2, –3) D' (–2, –3) C' D' Answer: The x-coordinates are opposite, but the y-coordinates stay the same. That is, (a, b) (–a, b). COORDINATE GEOMETRY Suppose quadrilateral ABCD with A(1, 2), B(3, 5), C(4, –3), and D(2, –5) is reflected in the origin. Graph ABCD and its image under reflection in the origin. Compare the coordinates of each vertex with the coordinates of its image. Use the horizontal and vertical distances from each vertex to the origin to find the coordinates of its image. From A to the origin is 2 units down and 1 unit left. A' is located by repeating that pattern from the origin. A(1, 2) A' (–1, –2) B(3, 5) B' (–3, –5) D' C' C(4, –3) C' (–4, 3) D(2, –5) D' (–2, 5) Plot the reflected vertices and connect to form the image A'B'C'D'. Comparing coordinates shows that (a, b) (–a, –b). Answer: Both the x- and y-coordinates are opposite. That is, (a, b) (–a, –b). A' B' COORDINATE GEOMETRY Suppose quadrilateral ABCD with A(1, 2), B(3, 5), C(4, –3), and D(2, –5) is reflected in the line y = x. Graph ABCD and its image under reflection in the line y = x. Compare the coordinates of each vertex with the coordinates of its image. The slope of y = x is 1. AA’ is perpendicular to y = x so its slope is –1. From A to the line y = x move down ½ unit and right ½ unit. From the line y = x move down ½ unit, right ½ unit to A'. C' A(1, 2) A'(2, 1) B(3, 5) B'(5, 3) C(4, –3) C'(–3, 4) B' D' A' D(2, –5) D'(–5, 2) Plot the reflected vertices and connect to form the image A'B'C'D'. Answer: The x-coordinate becomes the y-coordinate and the ycoordinate becomes the x-coordinate. That is, (a, b) (b, a). BILLARDS Dave challenged Juan to hit the 8 ball in the left corner pocket. Describe how Juan should hit the ball using reflections. Answer: Juan should mentally reflect the left corner pocket in the line that contains the right side of the table. If he hits the ball at the reflected image of the pocket, the ball will strike the right side and rebound on a path toward the left corner pocket. Determine how many lines of symmetry a regular pentagon has. Then determine whether a regular pentagon has a point of symmetry. A regular pentagon has five lines of symmetry. A point of symmetry is a point that is a common point of reflection for all points on the figure. There is not one point of symmetry in a regular pentagon. Answer: 5; no a. Determine how many lines of symmetry an equilateral triangle has. Then determine whether an equilateral triangle has a point of symmetry. Answer: 3; no b. Determine how many lines of symmetry a hexagon has. Then determine whether a hexagon has a point of symmetry. Answer: 6; yes Summary & Homework • Summary: Reflection x-axis y-axis origin y=x Pre-image to (a, b) (a, -b) (a, b) (-a, b) (a, b) (-a, -b) (a, b) (b, a) image Find Multiply y Multiply x Multiply both Interchange coordinates coordinate coordinate coordinates by -1 x and y by -1 by -1 coordinates – Line of Symmetry – a line across which the figure could be folded in half – Point of Symmetry – even numbered regular figures only for us • Homework: – Take notes on this PPT (Power Point )