1 / 58

The Distance and Midpoint Formulas

The Distance and Midpoint Formulas. Goal 1 Find the Midpoint of a Segment Goal 2 Find the distance between two points on a coordinate plane. Find the slope of a line between two points on a coordinate plane. Goal 3. Distance Formula. Used to find the distance between two points.

claudiarussell
Download Presentation

The Distance and Midpoint Formulas

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

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

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

  3. Example x1 y1 x2 y2 • 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 -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. Answer!

  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. (6, 5) "How steep am I?" Use the slope formula ( -5, -3) = =

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

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

  13. The Midpoint Formula The midpoint between the two points (x1, y1) and (x2, y2) is:

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

  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 (x1, y1) and (x2, y2) is given by:

  18. Find the Distance Between the points. Example 4 • (-2, 5) and (3, -1) • Let (x1, y1) = (-2, 5) and (x2, y2) = (3, -1)

  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). Homework 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: (x1, y1) and (x2, y2)

  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. Negative Positive Skiing Uphill Examples: Skiing Downhill Examples:

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

  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) Slope: 3 Classification: Positive Slope: 0 Classification: Zero 3. (-2, -1) and (-1, -4) 4. (2, 3) and (2, 7) Slope: -3 Classification: Negative Slope: Undefined Classification: Undefined 5. (-1, -1) and (5, 5) 6. (8, 4) and (6, 4) Slope: 1 Classification: Positive Slope: 0 Classification: Zero

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

  36. The Coordinate Plane Definition: 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. y - axis Origin x - axis Lesson 1-3: Formulas

  37. d = d = d = The Distance Formula Find the distance between (-3, 2) and (4, 1) The distance d between any two points with coordinates and is given by the formula d = . Example: x1 = -3, x2 = 4, y1 = 2 , y2 = 1 Lesson 1-3: Formulas

  38. M = M = Midpoint Formula In the coordinate plane, the coordinates of the midpoint of a segment whose endpoints have coordinates and are . Find the midpoint between (-2, 5) and (6, 4) Example: x1= -2, x2= 6, y1 = 5, and y2 = 4 Lesson 1-3: Formulas

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

  40. 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. Lesson 1-3: Formulas

  41. m = m = 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. • Find the slope between (3,4) and (3,-2) and describe the line. Since the slope is undefined, the line must be vertical. 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) left 11 (-11) y Solution: up 0 m (– 4, 6) (7, 6) x This line is horizontal. 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) right 0 y Solution: (– 3, 8) m up 10 x (– 3, – 2) undefined This line is vertical. Lesson 1-3: Formulas

  44. 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. Lesson 1-3: Formulas

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

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

  47. Evaluate (–14)2and (–6)2. Add 196 and 36. Take the square root. Example 6-1b 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, ifnecessary. 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. Evaluate powers. Simplify. Distance Formula Simplify. Example 6-2b

More Related