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Reflections

Reflections. Draw reflected images. Recognize and draw lines of symmetry and points of symmetry. Port Orchard by Ken Duffin. DRAW REFLECTIONS. Reflections are everywhere. What do you notice?. Every point is the same distance from the centerline, and

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Reflections

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  1. Reflections • Draw reflected images. • Recognize and draw lines of symmetry and points of • symmetry Port Orchard by Ken Duffin

  2. DRAW REFLECTIONS Reflections are everywhere. What do you notice? • Every point is the same distance from the centerline, and • The reflection is the same size as the original image.

  3. DRAW REFLECTIONS A reflection is a transformation representing the flip of a figure. D B C preimage A E m E’ A’ image C’ B’ D’ The line m is called the line of reflection for ABCDE and A’B’C’D’E’.

  4. DRAW REFLECTIONS It doesn’t matter what direction the line of reflection goes. The reflected image is always the same size. It just faces the other way.

  5. Y W U' X V' P Z' Z V X' U W' Y' DRAW REFLECTIONS It’s also possible to reflect a preimage in a point. In the figure below, polygon UVWXYZ is reflected in a point P. Note that P is the midpoint of each segment connecting a point with its image. When reflecting an image in a line or in a point, the image is congruent to the preimage. Thus, an image is a congruence transformation or an isometry.

  6. Example 1Reflecting a Figure in a Line Draw the reflected image of quadrilateral DEFG in line m. F m Step 1 Since D is on line m, D is its own reflection. Draw segments perpendicular to line m from E, F, and G. G E D

  7. Example 1Reflecting a Figure in a Line Draw the reflected image of quadrilateral DEFG in line m. F Step 2 Locate E’, F’ and G’ so that m is the perpendicular bisector of EE’, FF’, and GG’. Points E’, F’ and G’ are the respective images of E, F, and G. G G’ E F’ D E’

  8. Example 1Reflecting a Figure in a Line Draw the reflected image of quadrilateral DEFG in line m. F Step 3 Connect vertices D,E’, F’ and G’. G G’ E F’ D E’

  9. Example 2Reflecting a Figure in a Line Draw the reflected image of quadrilateral DEFG in line m, this time with point G on the opposite side of the line. F Step 1 Since D is on line m, D is its own reflection. Draw segments perpendicular to line m from E, F, and G. E G D

  10. Example 2Reflecting a Figure in a Line Draw the reflected image of quadrilateral DEFG in line m, this time with point G on the opposite side of the line. F Step 2 Locate E’, F’ and G’ so that m is the perpendicular bisector of EE’, FF’, and GG’. Points E’, F’ and G’ are the respective images of E, F, and G. G’ E G F’ D E’

  11. Example 2Reflecting a Figure in a Line Draw the reflected image of quadrilateral DEFG in line m, this time with point G on the opposite side of the line. F Step 3 Connect vertices D,E’, F’ and G’. G’ E G F’ D E’

  12. Example 3Reflections in the x-axis Quadrilateral KLMN has vertices K(2, -4), L(-1, 3), M(-4, 2), and N(-3, -4). Compare the coordinates of each vertex with the coordinates of its image reflected in the x-axis. Use vertical gridlines to find a corresponding point for each vertex in the reflection. 4 L 3 M 2 1 -8 -6 -4 -2 2 4 6 -1 -2 -3 -4 K N -5

  13. Example 3Reflections in the x-axis Quadrilateral KLMN has vertices K(2, -4), L(-1, 3), M(-4, 2), and N(-3, -4). Compare the coordinates of each vertex with the coordinates of its image reflected in the x-axis. Use vertical gridlines to find a corresponding point for each vertex in the reflection. N’ K’ 4 L 3 M 2 K(2, -4) → K’(2, 4) L(-1, 3) → L’(-1, -3) M(-4, 2)→ M’(-4, -2) N(-3, -4)→ N’(-3, 4) 1 -8 -6 -4 -2 2 4 6 -1 M’ -2 -3 L’ -4 K N -5

  14. Example 4Reflections in the y-axis Quadrilateral KLMN has vertices K(2, -4), L(-1, 3), M(-4, 2), and N(-3, -4). Compare the coordinates of each vertex with the coordinates of its image reflected in the y-axis. Use horizontal gridlines to find a corresponding point for each vertex in the reflection. 4 L 3 M 2 1 -8 -6 -4 -2 2 4 6 -1 -2 -3 -4 K N -5

  15. Example 4Reflections in the y-axis Quadrilateral KLMN has vertices K(2, -4), L(-1, 3), M(-4, 2), and N(-3, -4). Compare the coordinates of each vertex with the coordinates of its image reflected in the y-axis. Use horizontal gridlines to find a corresponding point for each vertex in the reflection. 4 L L’ 3 M M’ 2 K(2, -4) → K’(-2, -4) L(-1, 3) → L’(1, 3) M(-4, 2)→ M’(4, 2) N(-3, -4)→ N’(3, -4) 1 -8 -6 -4 -2 2 4 6 -1 -2 -3 N’ -4 K N K’ -5

  16. Example 5Reflections in the origin Quadrilateral KLMN has vertices K(2, -4), L(-1, 3), M(-4, 2), and N(-3, -4). Compare the coordinates of each vertex with the coordinates of its image reflected in the origin. Use horizontal and vertical distances to find a corresponding point for each vertex in the reflection. 4 L 3 M 2 1 -8 -6 -4 -2 2 4 6 -1 -2 -3 -4 K N -5

  17. Example 5Reflections in the origin Quadrilateral KLMN has vertices K(2, -4), L(-1, 3), M(-4, 2), and N(-3, -4). Compare the coordinates of each vertex with the coordinates of its image reflected in the origin. Use horizontal and vertical distances to find a corresponding point for each vertex in the reflection. N’ K’ 4 L 3 M 2 1 K(2, -4) → K’(-2, 4) L(-1, 3) → L’(1, -3) M(-4, 2)→ M’(4, -2) N(-3, -4)→ N’(3, 4) -8 -6 -4 -2 2 4 6 -1 M’ -2 -3 L’ -4 K N -5

  18. Example 5Reflections in the line x=y Quadrilateral KLMN has vertices K(2, -4), L(-1, 3), M(-4, 2), and N(-3, -4). Compare the coordinates of each vertex with the coordinates of its image reflected in the line x = y. Reflect the coordinates of each vertex in the line x = y 4 L 3 M 2 1 -8 -6 -4 -2 2 4 6 -1 -2 -3 -4 K N -5

  19. Example 5Reflections in the line x = y Quadrilateral KLMN has vertices K(2, -4), L(-1, 3), M(-4, 2), and N(-3, -4). Compare the coordinates of each vertex with the coordinates of its image reflected in the line x = y. Reflect the coordinates of each vertex in the line x = y 4 L K’ 3 M 2 1 K(2, -4) → K’(-4, 2) L(-1, 3) → L’(3, -1) M(-4, 2)→ M’(2, -4) N(-3, -4)→ N’(-4, -3) -8 -6 -4 -2 2 4 6 L’ -1 -2 N’ -3 M’ -4 K N -5

  20. Concept SummaryReflections in the Coordinate Plane

  21. Pool Table Problems The incoming angle equals the outgoing angle.

  22. Example 6Use Reflections Where do we aim on the bottom cushion so white hits blue?

  23. Example 7Use Reflections Where do we aim on the top cushion so white hits blue?

  24. Example 8Use Reflections Where do we aim on the left cushion so white hits the bottom cushion and then hits blue?

  25. LINES AND POINTS OF SYMMETRY Lines of Symmetry One Two More Than Two None F Points of Symmetry

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