Transformations and the Coordinate Plane

1. Transformations and the Coordinate Plane Eleanor Roosevelt High School Geometry Mr. Chin-Sung Lin

2. ERHS Math Geometry The Coordinates of a Point in a Plane Mr. Chin-Sung Lin

3. ERHS Math Geometry Two intersecting lines determine a plane. The coordinate plane is determined by a horizontal line, the x-axis, and a vertical line, the y-axis, which are perpendicular and intersect at a point called the origin Coordinate Plane Y X O Mr. Chin-Sung Lin

4. ERHS Math Geometry Every point on a plane can be described by two numbers, called the coordinates of the point, usually written as an ordered pair (x, y) Coordinate Plane Y (x, y) X O Mr. Chin-Sung Lin

5. ERHS Math Geometry The x-coordinate or the abscissa, is the distance from the point to the y-axis. The y-coordinate or the ordinate is the distance from the point to the x-axis. Point O, the origin, has the coordinates (0, 0) Coordinate Plane Y (x, y) y X O (0, 0) x Mr. Chin-Sung Lin

6. ERHS Math Geometry Two points are on the same horizontal line if and only if they have the same y-coordinates Postulates of Coordinate Plane Y (x1, y) (x2, y) X O Mr. Chin-Sung Lin

7. ERHS Math Geometry The length of a horizontal line segment is the absolute value of the difference of the x-coordinates d = |x2 – x1| Postulates of Coordinate Plane Y (x1, y) (x2, y) X O Mr. Chin-Sung Lin

8. ERHS Math Geometry Two points are on the same vertical line if and only if they have the same x-coordinates Postulates of Coordinate Plane Y (x, y2) (x, y1) X O Mr. Chin-Sung Lin

9. ERHS Math Geometry The length of a vertical line segment is the absolute value of the difference of the y-coordinates d = |y2 – y1| Postulates of Coordinate Plane Y (x, y2) (x, y1) X O Mr. Chin-Sung Lin

10. ERHS Math Geometry Each vertical line is perpendicular to each horizontal line Postulates of Coordinate Plane Y X O Mr. Chin-Sung Lin

11. ERHS Math Geometry From the origin, move to the right if the x-coordinate is positive or to the left if the x-coordinate is negative. If it is 0, there is no movement From the point on the x-axis, move up if the y-coordinate is positive or down if the y-coordinate is negative. If it is 0, there is no movement Locating a Point in the Coordinate Plane Y (x, y) y X O x Mr. Chin-Sung Lin

12. ERHS Math Geometry From the point, move along a vertical line to the x-axis.The number on the x-axis is the x-coordinate of the point From the point, move along a horizontal line to the y-axis.The number on the y-axis is the y-coordinate of the point Finding the Coordinates of a Point Y (x, y) y X O x Mr. Chin-Sung Lin

13. ERHS Math Geometry Graph the following points: A(4, 1), B(1, 5), C(-2,1). Then draw ∆ ABC and find its area Graphing on the Coordinate Plane Y X O Mr. Chin-Sung Lin

14. ERHS Math Geometry Graph the following points: A(4, 1), B(1, 5), C(-2,1). Then draw ∆ ABC and find its area Graphing on the Coordinate Plane Y B (1, 5) A (4, 1) C (-2, 1) D (1, 1) X O Mr. Chin-Sung Lin

15. ERHS Math Geometry • Graph the following points: A(4, 1), B(1, 5), C(-2,1). Then draw ∆ ABC and find its area • AC = | 4 – (-2) | = 6 • BD = | 5 – 1 | = 4 • Area = ½ (AC)(BD) • = ½ (6)(4) = 12 Graphing on the Coordinate Plane Y B (1, 5) A (4, 1) C (-2, 1) D (1, 1) X O Mr. Chin-Sung Lin

16. ERHS Math Geometry Line Reflections Mr. Chin-Sung Lin

17. ERHS Math Geometry Line Reflections Mr. Chin-Sung Lin

18. ERHS Math Geometry Line Reflections Line Reflection (Object & Image) Y Line of Reflection Mr. Chin-Sung Lin

19. ERHS Math Geometry Aone-to-one correspondence between two sets of points, S and S’, such that every point in set S corresponds to one and only one point in set S’, called its image, and every point in S’ is the image of one and only one point in S, called its preimage Transformation S’ S Mr. Chin-Sung Lin

20. ERHS Math Geometry If point P is not on k, then the image of P is P’ where k is the perpendicular bisector of PP’ If point P is on k, the image of P is P A Reflection in Line k P P’ P k Mr. Chin-Sung Lin

21. ERHS Math Geometry Under a line reflection, distance is preserved Given: Under a reflection in line k, the image of A is A’ and the image of B is B’ Prove: AB = A’B’ Theorem of Line Reflection - Distance A’ A B’ B k Mr. Chin-Sung Lin

22. ERHS Math Geometry Under a line reflection, distance is preserved Given: Under a reflection in line k, the image of A is A’ and the image of B is B’ Prove: AB = A’B’ Theorem of Line Reflection - Distance A’ A C B’ B D k Mr. Chin-Sung Lin

23. ERHS Math Geometry Under a line reflection, distance is preserved Given: Under a reflection in line k, the image of A is A’ and the image of B is B’ Prove: AB = A’B’ Theorem of Line Reflection - Distance A’ A C SAS B’ B D k Mr. Chin-Sung Lin

24. ERHS Math Geometry Under a line reflection, distance is preserved Given: Under a reflection in line k, the image of A is A’ and the image of B is B’ Prove: AB = A’B’ Theorem of Line Reflection - Distance A’ A C CPCTC B’ B D k Mr. Chin-Sung Lin

25. ERHS Math Geometry Under a line reflection, distance is preserved Given: Under a reflection in line k, the image of A is A’ and the image of B is B’ Prove: AB = A’B’ Theorem of Line Reflection - Distance A’ A C SAS B’ B D k Mr. Chin-Sung Lin

26. ERHS Math Geometry Under a line reflection, distance is preserved Given: Under a reflection in line k, the image of A is A’ and the image of B is B’ Prove: AB = A’B’ Theorem of Line Reflection - Distance A’ A C CPCTC B’ B D k Mr. Chin-Sung Lin

27. ERHS Math Geometry Since distance is preserved under a line reflection, the image of a triangle is a congruent triangle Theorem of Line Reflection - Distance A’ A M M’ SSS B’ B D’ C C’ D k Mr. Chin-Sung Lin

28. ERHS Math Geometry Under a line reflection, angle measure is preserved Under a line reflection, collinearity is preserved Under a line reflection, midpoint is preserved Corollaries of Line Reflection A’ A M M’ B’ B D’ C C’ D k Mr. Chin-Sung Lin

29. ERHS Math Geometry We use rk as a symbol for the image under a reflection in line k rk (A) = A’ rk (∆ ABC ) = ∆ A’B’C’ Notation of Line Reflection A’ A B’ B C C’ k Mr. Chin-Sung Lin

30. ERHS Math Geometry Ifrk (AC) = A’C’, construct A’C’ Construction of Line Reflection A C k Mr. Chin-Sung Lin

31. ERHS Math Geometry Construct the perpendicular line from A to k. Let the point of intersection be M Construction of Line Reflection M A C k Mr. Chin-Sung Lin

32. ERHS Math Geometry Construct the perpendicular line from C to k. Let the point of intersection be N Construction of Line Reflection M A N C k Mr. Chin-Sung Lin

33. ERHS Math Geometry Construct A’ on AM such that AM = A’M Construct C’ on CN such that CN = C’N Construction of Line Reflection A’ M A N C C’ k Mr. Chin-Sung Lin

34. ERHS Math Geometry Draw A’C’ Construction of Line Reflection A’ M A N C C’ k Mr. Chin-Sung Lin

35. ERHS Math Geometry Line Symmetry in Nature Mr. Chin-Sung Lin

36. ERHS Math Geometry A figure has line symmetry when the figure is its own image under a line reflection This line of reflection is a line of symmetry, or an axis of symmetry Line Symmetry Mr. Chin-Sung Lin

37. ERHS Math Geometry It is possible for a figure to have more than one axis of symmetry Line Symmetry Mr. Chin-Sung Lin

38. ERHS Math Geometry Line Reflections in the Coordinate Plane Mr. Chin-Sung Lin

39. ERHS Math Geometry Under a reflection in the y-axis, the image of P(a, b) is P’(-a, b) Reflection in the y-axis y Q(0, b) P(a, b) P’(-a, b) x O Mr. Chin-Sung Lin

40. ERHS Math Geometry If ∆ ABC is reflected in the y-axis, where A(-3, 3), B(-4, 1), and C(-1, 1), draw ry-axis (∆ ABC ) = ∆ A’B’C’ Reflection in the y-axis A(-3, 3) y B(-4, 1) C(-1, 1) x O Mr. Chin-Sung Lin

41. ERHS Math Geometry If ∆ ABC is reflected in the y-axis, where A(-3, 3), B(-4, 1), and C(-1, 1), draw ry-axis (∆ ABC ) = ∆ A’B’C’ Reflection in the y-axis A’(3, 3) A(-3, 3) y B’(4, 1) B(-4, 1) C(-1, 1) C’(1, 1) x O Mr. Chin-Sung Lin

42. ERHS Math Geometry Under a reflection in the x-axis, the image of P(a, b) is P’(a, -b) Reflection in the x-axis y P(a, b) Q(a, 0) x O P’(a, -b) Mr. Chin-Sung Lin

43. ERHS Math Geometry If ∆ ABC is reflected in the x-axis, where A(3, 3), B(4, 1), and C(1, 1), draw rx-axis (∆ ABC ) = ∆ A’B’C’ Reflection in the x-axis y A(3, 3) B(4, 1) C(1, 1) x O Mr. Chin-Sung Lin

44. ERHS Math Geometry If ∆ ABC is reflected in the x-axis, where A(3, 3), B(4, 1), and C(1, 1), draw rx-axis (∆ ABC ) = ∆ A’B’C’ Reflection in the x-axis y A(3, 3) B(4, 1) C(1, 1) x C’(1, -1) B’(4, -1) O A’(3, -3) Mr. Chin-Sung Lin

45. ERHS Math Geometry Under a reflection in the y =x, the image of P(a, b) is P’(b, a) Reflection in the Line y = x P(a, b) y R(b, b) P’(b, a) Q(a, a) O x Mr. Chin-Sung Lin

46. ERHS Math Geometry If ∆ ABC is reflected in the x-axis, where A(2, 2), B(1, 4), and C(-1, 1), draw ry=x (∆ ABC ) = ∆ A’B’C’ Reflection in the Line y = x B(1, 4) y A(2, 2) C(-1, 1) O x Mr. Chin-Sung Lin

47. ERHS Math Geometry If ∆ ABC is reflected in the x-axis, where A(2, 2), B(1, 4), and C(-1, 1), draw ry=x (∆ ABC ) = ∆ A’B’C’ * Point A is a fixed point since it is on the line of reflection Reflection in the Line y = x B(1, 4) y A(2, 2)=A’(2, 2) C(-1, 1) B’(4, 1) O x C’(1, -1) Mr. Chin-Sung Lin

48. ERHS Math Geometry Point Reflections in the Coordinate Plane Mr. Chin-Sung Lin

49. ERHS Math Geometry If point A is not point P, then the image of A is A’ and P the midpoint of AA’ The point P is its own image A Point Reflection in P y A P x O A’ Mr. Chin-Sung Lin

50. ERHS Math Geometry Under a point reflection, distance is preserved Theorem of Point Reflections y B’ A P x O A’ B Mr. Chin-Sung Lin