1. Goal 1 Find the Midpoint of a Segment Goal 2 Find the distance between two points on a coordinate plane Goal 3 Find the slope of a line between two points on a coordinate plane

2. Distance Formula  Used to find the distance between two points

3.   Find the distance between (2,1) and (5,2).  D= (2 - 5)² + (1 - 2)²  D= (-3)² + (-1)²  D= 9+1  D= 10  D= 3.162 Example x1y1x2y2 -First substitute numbers for variables and solve the parentheses. -Then solve the squared number. -Add the two numbers. -Find the square root of the remaining number.

4. Example  Find the distance between A(4,8) and B(1,12) A (4, 8)B (1, 12)

5. YOU TRY!!  Find the distance between:  A. (2, 7) and (11, 9)  B. (-5, 8) and (2, - 4)

6. Midpoint Formula  Used to find the center of a line segment

7. Example  Find the midpoint between A(4,8) and B(1,12) A (4, 8)B (1, 12)

8. YOU TRY!!  Find the midpoint between:  A) (2, 7) and (14, 9)  B) (-5, 8) and (2, - 4)

9.  THE SLOPE FORMULA!

10. ( -5, -3) (6, 5) Use the slope formula ==

11.   Complete the handout given in class. It is also posted on GradeSpeed and my website. Homework

12. Midpoint and Distance Formulas Goal 1 Find the Midpoint of a Segment Goal 2 Find the Distance Between Two Points on a Coordinate Plane 12.6

13. The Midpoint Formula The midpoint between the two points (x 1, y 1 ) and (x 2, y 2 ) is:

14. Find the midpoint of the segment whose endpoints are (6,-2) & (2,- 9) Example 1

15. Example 2 Find the coordinates of the midpoint of the segment whose endpoints are (5, 2) and ( 7, 8)

16. Example 3 Find the coordinates of the midpoint of the segment whose endpoints are (-2, 8) and ( 4, 0)

17. Distance Formula The distance between two points with coordinates (x 1, y 1 ) and (x 2, y 2 ) is given by:

18. Find the Distance Between the points. (-2, 5) and (3, -1) (-2, 5) and (3, -1) Let (x 1, y 1 ) = (-2, 5) and (x 2, y 2 ) = (3, -1) Let (x 1, y 1 ) = (-2, 5) and (x 2, y 2 ) = (3, -1) Example 4

19. Example 5 What is the distance between P(- 1, 4) and Q(2, - 3)?

20. Example 6 What is the distance between P(3, 0) and Q(5, - 4)?

21. Example 7 What is the distance between P(-5, 2) and Q(2, - 5)?

22. Example 8 Use the distance formula to determine whether the three points are vertices of a right triangle: (1,1), (4,4), (4,1)

23. Example 9 Use the distance formula to determine whether the three points are vertices of a right triangle: (3, -4), (-2, -1), (4, 6). p. 748 #16-28e, 36-44e, 61-63

24. There are formulas that you will be provided with to calculate various pieces of information about pairs of points. Each formula refers to a set of two points: (x 1, y 1 ) and (x 2, y 2 )

25. Distance – the length of the line segment that connects two given points in the coordinate plane. Distance Formula:

26. Ex#1: (2, 2) and (5, -2) Distance: ________

27. The midpoint is the point equidistant between two points in the coordinate plane. Midpoint Formula: NOTICE: the answer to a midpoint formula problem will be in the form (1, 2) – meaning your answer is another point!

28. Ex# 1: (2, 2) and (5, -2) Midpoint: _______

29. The slope is the ratio of vertical change (rise) to horizontal change (run) of a line. Slope Formula:

30. Ex# 1: (2, 2) and (5, -2) Slope: __________

31. Example 2: (0, 3) and (-1, 1) Distance: ________ Midpoint: _______ Slope: __________

32. Slope: There are four classifications of slope: positive, negative, zero, and undefined. Positive Negative Skiing Uphill Examples: Skiing Downhill Examples:

33. Slope: There are four classifications of slope: positive, negative, zero, and undefined. Zero Undefined Cross Country Skiing Examples: If you tried to ski on this, you wouldn’t make it. Examples: You have an undefined slope whenever you get a zero in the denominator.

34. Independent Practice: Calculate the slope for each pair of points. Classify each slope as positive, negative, zero, or undefined. 1. (2, 2) and (3, 5)2. (0, 0) and (3, 0) 3. (-2, -1) and (-1, -4)4. (2, 3) and (2, 7) 5. (-1, -1) and (5, 5)6. (8, 4) and (6, 4) Slope: 3 Classification: Positive Slope: -3 Classification: Negative Slope: 1 Classification: Positive Slope: 0 Classification: Zero Slope: Undefined Classification: Undefined Slope: 0 Classification: Zero

35.  Lesson 1-3: Formulas35 Lesson 1-3 Formulas

36.  Lesson 1-3: Formulas36 The Coordinate Plane In the coordinate plane, the horizontal number line (called the x- axis) and the vertical number line (called the y- axis) interest at their zero points called the Origin. Definition: x - axis y - axis Origin

37. Lesson 1-3: Formulas37 The Distance Formula Find the distance between (-3, 2) and (4, 1) x 1 = -3, x 2 = 4, y 1 = 2, y 2 = 1 d = The distance d between any two points with coordinates and is given by the formula d =. Example:

38. Lesson 1-3: Formulas 38 Midpoint Formula M = Find the midpoint between (-2, 5) and (6, 4) x 1 = -2, x 2 = 6, y 1 = 5, and y 2 = 4 Example: In the coordinate plane, the coordinates of the midpoint of a segment whose endpoints have coordinates and are.

39. Lesson 1-3: Formulas 39 Slope Formula Find the slope between (-2, -1) and (4, 5).Example: Definition:In a coordinate plane, the slope of a line is the ratio of its vertical rise over its horizontal run. Formula:The slope m of a line containing two points with coordinates and is given by the formula where.

40.  Lesson 1-3: Formulas40 Describing Lines  Lines that have a positive slope rise from left to right.  Lines that have a negative slope fall from left to right.  Lines that have no slope (the slope is undefined) are vertical.  Lines that have a slope equal to zero are horizontal.

41.  Lesson 1-3: Formulas41 Some More Examples  Find the slope between (4, -5) and (3, -5) and describe it. Since the slope is zero, the line must be horizontal. m = Find the slope between (3,4) and (3,-2) and describe the line. m = Since the slope is undefined, the line must be vertical.

42. Lesson 1-3: Formulas 42 Example 3 : Find the slope of the line through the given points and describe the line. (7, 6) and (– 4, 6) Solution: This line is horizontal. x y (7, 6) up 0 left 11 (-11) (– 4, 6) m

43. Lesson 1-3: Formulas 43 Example 4: Find the slope of the line through the given points and describe the line. (– 3, – 2) and (– 3, 8) Solution: This line is vertical. x y (– 3, – 2) up 10 right 0 (– 3, 8) undefined m

44. Lesson 1-3: Formulas44 Practice Find the distance between (3, 2) and (-1, 6). Find the midpoint between (7, -2) and (-4, 8). Find the slope between (-3, -1) and (5, 8) and describe the line. Find the slope between (4, 7) and (-4, 5) and describe the line. Find the slope between (6, 5) and (-3, 5) and describe the line.

45. Lesson 6 Contents Example 1Use the Distance Formula Example 2Use the Distance Formula to Solve a Problem Example 3Use the Midpoint Formula

46. Example 6-1a Find the distance between M(8, 4) and N(–6, –2). Round to the nearest tenth, if necessary. Use the Distance Formula. Distance Formula Simplify.

47. Example 6-1b Evaluate (–14) 2 and (–6) 2. Add 196 and 36. Take the square root. Answer:The distance between points M and N is about 15.2 units.

48. Example 6-1c Find the distance between A(–4, 5) and B(3, –9). Round to the nearest tenth, if necessary. Answer:The distance between points A and B is about 15.7 units.

49. Example 6-2a Geometry Find the perimeter of  XYZ to the nearest tenth. First, use the Distance Formula to find the length of each side of the triangle.

50. Example 6-2b Distance Formula Simplify. Evaluate powers. Simplify.

51. Example 6-2c Distance Formula Simplify. Evaluate powers. Simplify.

52. Example 6-2d Distance Formula Simplify. Evaluate powers. Simplify.

53. Example 6-2e Then add the lengths of the sides to find the perimeter. Answer:The perimeter is about 15.8 units.

54. Example 6-2f Geometry Find the perimeter of  ABC to the nearest tenth. Answer:The perimeter is about 21.3 units.

55. Example 6-3a Find the coordinates of the midpoint of

56. Example 6-3b Substitution Simplify. Answer:The coordinates of the midpoint of are (3, 3). Midpoint Formula

57. Example 6-3c Find the coordinates of the midpoint of Answer:The coordinates of the midpoint of are (1, –1).

58. End of Lesson 6