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Vocabulary Reflection – is a transformation representing a flip of a figure; figure may be reflected in a point, a line, or a plane. Line of Reflection– line that the figure can be folded so that the two halves match exactly

Common reflections in the coordinate plane x-axis y-axis y = x y = - x Pre-image to image (x, y) (x, -y) (x, y)  (-x, y) (x, y)  (y, x) (x, y) (-y, -x) Find coordinates Multiply y coordinate by -1 Multiply x coordinate by -1 Interchange x and y coordinates Interchange x and y coordinates and multiply by -1 The line of reflection 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'. Example 1-1a

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) A' B' D(2, –3)  D' (2, 3) Answer: The x-coordinates stay the same, but the y-coordinates are opposite. That is, (x, y)  (x, –y). Example 1-2a

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) C' D(2, –3)  D' (–2, –3) D' Answer: The x-coordinates are opposite, but the y-coordinates stay the same. That is, (x, y)  (–x, y). Example 1-3a

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' B(3, 5)  B'(5, 3) D' C(4, –3)  C'(–3, 4) 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 y-coordinate becomes the x-coordinate. That is, (x, y)  (y, x). Example 1-5a

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 under reflection in the line y = - x. 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' (–2, –1) B(3, 5)  B' (–5, –3) C(4, –3)  C' (3, –4) D(2, –5)  D' (5, –2) A' D' Plot the reflected vertices and connect to form the image A'B'C'D'. Comparing coordinates shows that (a, b)  (–a, –b). B' C' Answer: Both the x- and y-coordinates are opposite. That is, (x, y)  (–y, –x). Example 1-4a

Summary & Homework Summary: Homework: Geometry – Reflections Homework Sheet, to be completed on Graph Paper Reflection x-axis y-axis y = x y = - x Pre-image to image (x, y) (x, -y) (x, y)  (-x, y) (x, y)  (y, x) (x, y) (-y, -x) Find coordinates Multiply y coordinate by -1 Multiply x coordinate by -1 Interchange x and y coordinates Interchange x and y coordinates and multiply by -1