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Co-ordinate Geometry 1 Contents 1.Distance between points (Simple)Distance between points (Simple) 2.Pythagoras and Distance between two pointsPythagoras and Distance between two points 3.The Distance FormulaThe Distance Formula 4.Midpoint FormulaMidpoint Formula 5.GradientGradient 6.Equations of straight lineEquations of straight line 7.Parallel LinesParallel Lines Press “ctrl-A” GeoGebra

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Co-ordinate Geometry 2 10.1 Distance between Two Points (1/5)1 2 3 4 56 7 89101112131415 0 1 2 3 4 5 6 7 -2 -3 Distance from (0,3) to (7,3) = 7

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Co-ordinate Geometry 3 10.1 Distance between Two Points (2/5)1 2 3 4 56 7 89101112131415 0 1 2 3 4 5 6 7 -2 -3 Distance from (7,-3) to (7,3) = 6

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Co-ordinate Geometry 4 10.1 Distance between Two Points (3/5)1 2 3 4 56 7 89101112131415 0 1 2 3 4 5 6 7 -2 -3 Distance from (2,5) to (7,5) = 5

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Co-ordinate Geometry 5 10.1 Distance between Two Points (4/5)1 2 3 4 56 7 89101112131415 0 1 2 3 4 5 6 7 -2 -3 Distance from (13,7) to (13,2) = 5

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Co-ordinate Geometry 6 10.1 Distance between Two Points (5/5)1 2 3 4 56 7 89101112131415 0 1 2 3 4 5 6 7 -2 -3 Distance from (3,-2) to (12,-2) = 9

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Co-ordinate Geometry 7 1 2 3 4 56 7 8 0 1 2 3 4 5 6 7 -2 -3 10.1 Distance – Pythagoras (1/3) Distance from (1,1) to (4,5) = 5 3 4 c 2 = a 2 + b 2 c 2 = 3 2 + 4 2 c 2 = 9 + 16 c 2 = 25 c = 5 GeoGebra

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Co-ordinate Geometry 8 1 2 3 4 56 7 8 0 1 2 3 4 5 6 7 -2 -3 10.1 Distance – Pythagoras (2/3) Distance from (2,2) to (7,6) = 41 5 4 c 2 = a 2 + b 2 c 2 = 5 2 + 4 2 c 2 = 25 + 16 c 2 = 41 c = 41

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Co-ordinate Geometry 9 1 2 3 4 56 7 8 0 1 2 3 4 5 6 7 -2 -3 10.1 Distance – Pythagoras (3/3) Distance from (-1,-2) to (5,5) = 85 6 7 c 2 = a 2 + b 2 c 2 = 6 2 + 7 2 c 2 = 36 + 49 c 2 = 85 c = 85

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Co-ordinate Geometry 10 1 2 3 4 56 7 8 0 1 2 3 4 5 6 7 -2 -3 10.1 Distance – Pythagoras (1/1) Distance from (1,-2) to (7,7) 6 9 c 2 = a 2 + b 2 c 2 = 6 2 + 9 2 c 2 = 36 + 81 c 2 = 117 c = 10.816 65 c ≈ 10.8 ≈ 10.8

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Co-ordinate Geometry 11 1 2 3 4 56 7 8 0 1 2 3 4 5 6 7 -2 -3 10.2 Distance Formula (1/3) Distance from (3,-2) to (7,5)? (3,-2) (7,5) d = (x 2 -x 1 )2 +(y 2 -y 1 ) 2 = (7 - 3) 2 +(5 - - 2) 2 (x 1,y 1 ) (x 2,y 2 ) = 4 2 + 7 2 = 16 + 49 = 65 (as surd) Exact ≈ 8.06 (approx) GeoGebra

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Co-ordinate Geometry 12 10.2 Distance Formula (2/3) Distance from (12,2) to (4,5)? d = (x 2 -x 1 ) 2 +(y 2 -y 1 ) 2 = (4 - 12) 2 +(5 -2) 2 (x 1,y 1 ) (x 2,y 2 ) = (-8) 2 + 3 2 = 64 + 9 = 73 (as surd) Exact

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Co-ordinate Geometry 13 10.2 Distance Formula (3/3) Distance from (6,2) to (7,5)? d = (x 2 -x 1 ) 2 +(y 2 -y 1 ) 2 = (7 - 6) 2 +(5 -2) 2 (x 1,y 1 ) (x 2,y 2 ) = 1 2 + 3 2 = 1 + 9 = 10 (2 decimal places) Approximate = 3.16

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Co-ordinate Geometry 14 1 2 3 4 56 7 8 0 1 2 3 4 5 6 7 -2 -3 10.3 Midpoint Formula (1/2) Midpoint of (3,-2) to (7,6)? (3,-2) (7,6) (x 1,y 1 ) (x 2,y 2 ) Midpoint M. M = ( x 1 +x 2 2 y 1 +y 2 2 ), = ( 3+7 2 -2+6 2 ), = (5, 2) GeoGebra

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Co-ordinate Geometry 15 10.3 Midpoint Formula (2/2) Midpoint of (12,2) to (4,5)? (x 1,y 1 ) (x 2,y 2 ) M = ( x 1 +x 2 2 y 1 +y 2 2 ), = ( 12+4 2 2+5 2 ), = (8, 3.5)

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Co-ordinate Geometry 16 4 7-3 10.4 Gradient (1/5)Gradient rate of change. is the rate of change. Gradient = Horizontal Run Vertical Rise 123456 1 2 3 4 5 6 7 (1,3) (5,7) 5-1 4 m m = rise run = 4444 = 1 GeoGebra

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Co-ordinate Geometry 17 -4 7-1 10.4 Gradient (2/5)123456 1 2 3 4 5 6 7 (5,1) (1,7) 1-5 6 m = rise run = 6 -4 = 3 -2 = -3 2 Wolfram Demo

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Co-ordinate Geometry 18 10.4 Gradient (3/5) Gradients Gradients can be: Positive Increasing An Increasing function ZeroHorizontal Negative Decreasing A Decreasing function

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Co-ordinate Geometry 19 10.4 Gradient Formula (4/5)123456 1 2 3 4 5 6 7 y y (x 2,y 2 ) (x 1,y 1 ) y 2 -y 1 x 2 -x 1 m = rise run = y 2 -y 1 x 2 -x 1

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Co-ordinate Geometry 20 10.4 Gradient Formula (5/5) Gradient Gradient of (4,2) to (8,10)? (x 1,y 1 ) (x 2,y 2 ) m = y 2 -y 1 x 2 -x 1 = 10-2 8-4 = 8484 = 2 Gradient Gradient of (5,9) to (7,3)? (x 1,y 1 ) (x 2,y 2 ) m = y 2 -y 1 x 2 -x 1 = 3-9 7-5 = -6 2 = -3

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

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

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Co-ordinate Geometry 23 10.5 Linear Equations (2/6) Write in Gradient-Intercept Form a) 3y = 9x + 6 ÷3÷3÷3 y = 3x + 2 b) 5y = 2x + 1 ÷5÷5÷5 y = x + 2515 c) y - 2x = 0 +2x+2x y = 2x d) 3y + x -1 = 0 -x -x+1 +1 3y = -x +1 y = x + 313 ÷3 ÷3 ÷3

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Co-ordinate Geometry 24 10.5 Linear Equations (3/6) Write the Gradient and Y-Intercept a) y = 9x + 6 b) y = x - 1 c) y = 2x d) y = x + 7 12 m = 9 b = 6 m = b = -1 12 m = 2 b = 0 m = 1 b = 7

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Co-ordinate Geometry 25 10.5 Linear Equations (4/6) Write the equation. y = mx + b a) m=2 b=1 b) m= b=-5 c) m=-4 b=-1 d) m=12 b=7 23 y = 2x + 1 y = x - 5 23 y = -4x - 1 y = 12x + 7

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Co-ordinate Geometry 26 y = x + 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 32 +2x+2x y = 2x + 1 m = 2 b = 1 ÷2÷2÷2 72 m = b = 32 72

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Co-ordinate Geometry 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) c) y – 2x = 1 (1,3) d) y = 5x + 1 (1,5) 5 = 2x2 + 1 5 = 5 Yes 24 = 31 + 7 2x4 = 3x1 + 7 8 = 10 No 3 - 2x1 = 1 1 = 1 Yes 5 = 5x1 + 1 5 = 6 No

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Co-ordinate Geometry 28 10.6 Parallel Lines (1/5) Are two lines parallel? Do they have the same gradient? m 1 = m 2 y = mx + b 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

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Co-ordinate Geometry 29 10.6 Parallel Lines (2/5) Are the two lines parallel? 10x + 2y - 7 = 0 5x + y - 3 = 0 2y - 7 = -10x 2y = -10x + 7 y = -5x + 3.5 y - 3 = -5x y = -5x + 3 m 1 = -5 m 2 = -5 They are parallel !

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Co-ordinate Geometry 30 10.6 Parallel Lines (3/5) Are the two lines parallel? 6x + 2y - 3 = 0 3x + y - 3 = 0 2y - 3 = -6x 2y = -6x + 3 y = -2x + 1.5 y - 3 = -3x y = -3x + 3 m 1 = -2 m 2 = -3 They are NOT parallel !

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Co-ordinate Geometry 31 10.6 Parallel Lines (4/5) Are the interval and line parallel? (2,3) to (5,9) 4x + 2y - 6 = 0 m = y 2 – y 1 x 2 - x 1 = 2y - 6 = -4x y = -2x + 1.5 m1 = 2m1 = 2m1 = 2m1 = 2 m 2 = -2 They are NOT parallel ! 4y = -8x + 6 9 – 3 5 - 2

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Co-ordinate Geometry 32 10.6 Parallel Lines (5/5) Are the interval and line parallel? (2,9) to (5,3) 6x + 3y - 5 = 0 m = y 2 – y 1 x 2 -x 1 = 3y - 5 = -6x y = -2x + 1.25 m 1 = -2 m 2 = -2 They are parallel ! 4y = -8x + 5 3 – 9 5 - 2

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