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Distance and Equations in Coordinate Geometry

Learn how to calculate distance between two points using the Pythagorean theorem and explore equations of straight lines. Understand the concept of parallel lines.

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Distance and Equations in Coordinate Geometry

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  1. Contents • Distance between points (Simple) • Pythagoras and Distance between two points • The Distance Formula • Midpoint Formula • Gradient • Equations of straight line • Parallel Lines GeoGebra Press “ctrl-A”

  2. 7 6 5 4 3 2 1 0 3 4 -1 2 5 6 8 9 10 11 12 13 14 15 1 7 -1 -2 -3 10.1 Distance between Two Points (1/5) = 7 to (7,3) Distance from (0,3)

  3. 7 6 5 4 3 2 1 0 3 4 -1 2 5 6 8 9 10 11 12 13 14 15 1 7 -1 -2 -3 10.1 Distance between Two Points (2/5) = 6 to (7,3) Distance from (7,-3)

  4. 7 6 5 4 3 2 1 0 3 4 -1 2 5 6 8 9 10 11 12 13 14 15 1 7 -1 -2 -3 10.1 Distance between Two Points (3/5) = 5 to (7,5) Distance from (2,5)

  5. 7 6 5 4 3 2 1 0 3 4 -1 2 5 6 8 9 10 11 12 13 14 15 1 7 -1 -2 -3 10.1 Distance between Two Points (4/5) = 5 to (13,2) Distance from (13,7)

  6. 7 6 5 4 3 2 1 0 3 4 -1 2 5 6 8 9 10 11 12 13 14 15 1 7 -1 -2 -3 10.1 Distance between Two Points (5/5) = 9 to (12,-2) Distance from (3,-2)

  7. 7 6 5 4 3 2 1 0 3 4 -1 2 5 6 8 1 7 -1 -2 -3 10.1 Distance – Pythagoras (1/3) c2 = a2 + b2 c2 = 32 + 42 4 c2 = 9 + 16 3 c2 = 25 c= 5 = 5 to (4,5) Distance from (1,1) GeoGebra

  8. 7 6 5 4 3 2 1 0 3 4 -1 2 5 6 8 1 7 -1 -2 -3 10.1 Distance – Pythagoras (2/3) c2 = a2 + b2 4 c2 = 52 + 42 c2 = 25 + 16 5 c2 = 41 c= 41 = 41 to (7,6) Distance from (2,2)

  9. 7 6 5 4 3 2 1 0 3 4 -1 2 5 6 8 1 7 -1 -2 -3 10.1 Distance – Pythagoras (3/3) c2 = a2 + b2 c2 = 62 + 72 c2 = 36 + 49 7 c2 = 85 c= 85 6 = 85 to (5,5) Distance from (-1,-2)

  10. 7 6 5 4 3 2 1 0 3 4 -1 2 5 6 8 1 7 -1 -2 -3 10.1 Distance – Pythagoras (1/1) c2 = a2 + b2 c2 = 62 + 92 9 c2 = 36 + 81 c2 = 117 c= 10.81665 6 c≈ 10.8 to (7,7) ≈ 10.8 Distance from (1,-2)

  11. 10.2 Distance Formula (1/3) d = (x2-x1)2+(y2-y1)2 7 (x2,y2) 6 (7,5) = (7 - 3)2+(5 --2)2 5 4 = 42 + 72 3 2 = 16+ 49 1 0 (as surd) Exact 3 4 -1 2 5 6 8 1 7 = 65 -1 -2 (3,-2) (approx) ≈ 8.06 (x1,y1) -3 to (7,5)? Distance from (3,-2) GeoGebra

  12. 10.2 Distance Formula (2/3) (x1,y1) (x2,y2) to (4,5)? Distance from (12,2) d = (x2-x1)2+(y2-y1)2 = (4 - 12)2+(5 -2)2 = (-8)2 + 32 = 64+ 9 (as surd) Exact = 73

  13. 10.2 Distance Formula (3/3) (x1,y1) (x2,y2) to (7,5)? Distance from (6,2) d = (x2-x1)2+(y2-y1)2 = (7 - 6)2+(5 -2)2 = 12 + 32 = 1+ 9 (2 decimal places) Approximate = 3.16 = 10

  14. 10.3 Midpoint Formula (1/2) (x2,y2) 7 , (7,6) ( ) 6 x1+x2 2 y1+y2 2 M = 5 Midpoint M. 4 , 3 ( ) 3+7 2 -2+6 2 = 2 1 0 3 4 -1 2 5 6 8 1 7 = (5, 2) -1 -2 (3,-2) (x1,y1) -3 to (7,6)? Midpoint of (3,-2) GeoGebra

  15. 10.3 Midpoint Formula (2/2) (x1,y1) (x2,y2) to (4,5)? Midpoint of (12,2) , ( ) x1+x2 2 y1+y2 2 M = , ( ) 12+4 2 2+5 2 = = (8, 3.5)

  16. 10.4 Gradient (1/5) Vertical Rise 7 6 5 4 3 2 1 1 2 3 4 5 6 Gradient is the rate of change. Gradient = Horizontal Run (5,7) rise run m = m 4 7-3 5-1 4 4 4 = (1,3) = 1 GeoGebra

  17. 10.4 Gradient (2/5) 7 6 5 4 3 2 1 1 2 3 4 5 6 rise run (1,7) m = 6 -4 = 6 7-1 3 -2 (5,1) = -4 1-5 -3 2 = Wolfram Demo

  18. 10.4 Gradient (3/5) Gradientscan be: Positive An Increasing function Zero Horizontal Negative A Decreasing function

  19. 10.4 Gradient Formula (4/5) 7 6 5 4 3 2 1 1 2 3 4 5 6 y (x2,y2) rise run m = y2-y1 y2-y1 = x2-x1 x2-x1 (x1,y1) y

  20. 10.4 Gradient Formula (5/5) (x1,y1) (x2,y2) to (8,10)? Gradient of (4,2) y2-y1 10-2 8 4 m = = = = 2 x2-x1 8-4 (x1,y1) (x2,y2) to (7,3)? Gradient of (5,9) y2-y1 3-9 -6 2 m = = = = -3 x2-x1 7-5

  21. 10.5 Linear Equations Two types of equation. 1. Gradient-Intercept Form y = mx +b Gradient y-Intercept 2. General Form ax +by + c = 0 ‘b’ NOT y-intercept ‘a’ always positive. Always ‘0’ We must be able to convert between forms.

  22. 10.5 Linear Equations (1/6) Write in General Form a) y = 2x + 3 b) 2y = x +7 -y -y -2y -2y 0 = 2x – y + 3 0 = x – 2y +7 2x –y +3 = 0 x – 2y +7 = 0 x 3 x3 d) y = +3 c) y = -x + 6 x3 +x -6 +x -6 x3 x + y - 6 = 0 3y = x + 9 -3y -3y 0 = x -3y + 9 x -3y + 9 = 0

  23. 10.5 Linear Equations (2/6) Write in Gradient-Intercept Form a) 3y = 9x + 6 b) 5y = 2x + 1 ÷3 ÷3 ÷3 ÷5 ÷5 ÷5 2 5 1 5 y = 3x + 2 y = x + c) y - 2x = 0 d) 3y + x -1 = 0 +1 +1 -x +2x +2x -x y = 2x 3y = -x +1 ÷3 ÷3 ÷3 -1 3 1 3 y = x +

  24. 10.5 Linear Equations (3/6) Write the Gradient and Y-Intercept 1 2 a) y = 9x + 6 b) y = x - 1 1 2 m = 9 m = b = 6 b = -1 c) y = 2x d) y = x + 7 m = 2 m = 1 b = 0 b = 7

  25. 10.5 Linear Equations (4/6) Write the equation. y = mx + b 2 3 a) m=2 b=1 b) m= b=-5 2 3 y = 2x + 1 y = x - 5 c) m=-4 b=-1 d) m=12 b=7 y = 12x + 7 y = -4x - 1

  26. 10.5 Linear Equations (5/6) Write the equation as y=mx+b and state m and b. a) y – 2x = 1 b) 2y = 3x + 7 +2x +2x ÷2 ÷2 ÷2 3 2 7 2 y = 2x + 1 y = x + m = 2 3 2 m = b = 1 7 2 b =

  27. 10.5 Linear Equations (6/6) Is the point given on the line? a) y = 2x + 1 (2,5) b) 2y = 3x + 7 (1,4) 5 = 2x2 + 1 2x4 = 3x1 + 7 Yes No 5 = 5 8 = 10 d) y = 5x + 1 (1,5) c)y – 2x = 1 (1,3) 3 - 2x1 = 1 5 = 5x1 + 1 1 = 1 Yes 5 = 6 No

  28. 10.6 Parallel Lines (1/5) Are two lines parallel? Do they have the same gradient? m1 = m2 To find out if two lines are parallel put in the form y=mx+b Check to see if coefficients of x are equal. GeoGebra

  29. 10.6 Parallel Lines (2/5) Are the two lines parallel? 5x + y - 3 = 0 10x + 2y - 7 = 0 y - 3 = -5x 2y - 7 = -10x y = -5x + 3 2y = -10x + 7 y = -5x + 3.5 m1=-5 m2=-5 They are parallel !

  30. 10.6 Parallel Lines (3/5) Are the two lines parallel? 3x + y - 3 = 0 6x + 2y - 3 = 0 y - 3 = -3x 2y - 3 = -6x y = -3x + 3 2y = -6x + 3 y = -2x + 1.5 m1=-2 m2=-3 They are NOT parallel !

  31. 10.6 Parallel Lines (4/5) Are the interval and line parallel? 4x + 2y - 6 = 0 (2,3) to (5,9) 2y - 6 = -4x y2 – y1 x2 - x1 m = 4y = -8x + 6 9 – 3 5 - 2 y = -2x + 1.5 = m2=-2 m1=2 They are NOT parallel !

  32. 10.6 Parallel Lines (5/5) Are the interval and line parallel? 6x + 3y - 5 = 0 (2,9) to (5,3) 3y - 5 = -6x y2 – y1 x2-x1 m = 4y = -8x + 5 3 – 9 5 - 2 y = -2x + 1.25 = m1=-2 m2=-2 They are parallel !

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