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Lesson 2.4 Curves and Circles pp. 54-59

Lesson 2.4 Curves and Circles pp. 54-59. Objectives: 1. To define a triangle and related terms. 2. To classify curves. 3. To define a circle and related terms. 4. To state the Jordan Curve Theorem. Definition.

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Lesson 2.4 Curves and Circles pp. 54-59

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  1. Lesson 2.4 Curves and Circlespp. 54-59

  2. Objectives: 1. To define a triangle and related terms. 2. To classify curves. 3. To define a circle and related terms. 4. To state the Jordan Curve Theorem.

  3. Definition A triangle is the union of segments that connect three noncollinear points. A triangle is designated by the symbol  followed by the three noncollinear points.

  4. Triangle R T S Denoted: RST

  5. R T RT RS ST S Triangle opposite sides

  6. Definition A closed curve is a curve that begins and ends at the same point.

  7. Definition A simple curve is a curve that does not intersect itself (unless the starting and ending points coincide).

  8. Definition A simple closed curve is a simple curve that is also a closed curve.

  9. Definition A circle is the set of all points that are a given distance from a given point in a given plane. The center of the circle is the given point in the plane.

  10. O

  11. Definition A radius of a circle is a segment that connects a point on the circle with the center. (The plural of radius is radii.) A chord of a circle is a segment having both endpoints on the circle.

  12. A O

  13. B C O

  14. Definition A diameter is a chord that passes through the center of the circle. An arc is a curve that is a subset of a circle. (symbol: )

  15. B C A O AE D E

  16. Definition The interior of a circle is the set of all planar points whose distance from the center of the circle is less than the length of the radius (r).

  17. O

  18. Definition The exterior of a circle is the set of all planar points whose distance from the center of the circle is greater than the length of the radius (r).

  19. O

  20. Theorem 2.1 Jordan Curve Theorem. Any simple closed curve divides a plane into three disjoint sets: the curve itself, its interior, and its exterior.

  21. Definition A region is the union of a simple closed curve and its interior. The curve is the boundary of the region.

  22. Homework pp. 58-59

  23. ►A. Exercises Classify each figure as (1) a curve, (2) a closed curve, (3) a simple curve, (4) a simple closed curve, (5) or not a curve. Use the most specific term possible. 11.

  24. ►A. Exercises Classify each figure as (1) a curve, (2) a closed curve, (3) a simple curve, (4) a simple closed curve, (5) or not a curve. Use the most specific term possible. 13.

  25. ►A. Exercises Classify each figure as (1) a curve, (2) a closed curve, (3) a simple curve, (4) a simple closed curve, (5) or not a curve. Use the most specific term possible. 15.

  26. ►A. Exercises Classify each figure as (1) a curve, (2) a closed curve, (3) a simple curve, (4) a simple closed curve, (5) or not a curve. Use the most specific term possible. 17.

  27. A B C D E ►B. Exercises Use the figure for exercises 18-22. 19. Name all the angles.

  28. A B C D E ►B. Exercises Use the figure for exercises 18-22. 21. ABD  ADE

  29. ►B. Exercises 23. If X, Y, and Z are noncollinear, find XY  YZ  XZ.

  30. ►B. Exercises Use the figure shown for exercises 25-29. S is the region bounded by rectangle AHFC. Tell whether the statements is true or false. A B C D I H G F E 25. BCI S = BCF

  31. ►B. Exercises Use the figure shown for exercises 25-29. S is the region bounded by rectangle AHFC. Tell whether the statements is true or false. A B C D I H G F E 27. S BGED = BGFC

  32. 29. ABGH BGFC = ACFH  BG ►B. Exercises Use the figure shown for exercises 25-29. S is the region bounded by rectangle AHFC. Tell whether the statements is true or false. A B C D I H G F E

  33. ■ Cumulative Review True/False 32. The intersection of two planes can be a single point.

  34. ■ Cumulative Review True/False 33. The intersection of two opposite half- planes is their common edge.

  35. ■ Cumulative Review True/False 34. A segment is a curve.

  36. ■ Cumulative Review True/False 35. The Line Separation Postulate asserts that a line separates a plane into three disjoint sets.

  37. ■ Cumulative Review True/False 36. If planes s and t are parallel, then every line in plane s is parallel to every line in plane t.

  38. Analytic Geometry Graphing Lines and Curves

  39. Graph y = x + 2 x y

  40. Graph y = -x2 x y

  41. Graph y = x x y

  42. 1. Graph y = x - 5 x y 0 -5 1 -4 2 -3 5 0

  43. 2. Graph y = 3x x y 0 0 1 3 -1 -3

  44. 3. Graph y = x2 + 1 x y 0 1 1 2 -1 2 2 5 -2 5

  45. 4. Graph y = 2x + 3 x y

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