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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 Distance between Two Points (1/5) Distance from (0,3) to (7,3) = 7

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Co-ordinate Geometry Distance between Two Points (2/5) Distance from (7,-3) to (7,3) = 6

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Co-ordinate Geometry Distance between Two Points (3/5) Distance from (2,5) to (7,5) = 5

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Co-ordinate Geometry Distance between Two Points (4/5) Distance from (13,7) to (13,2) = 5

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Co-ordinate Geometry Distance between Two Points (5/5) Distance from (3,-2) to (12,-2) = 9

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Co-ordinate Geometry Distance – Pythagoras (1/3) Distance from (1,1) to (4,5) = c 2 = a 2 + b 2 c 2 = c 2 = c 2 = 25 c = 5 GeoGebra

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Co-ordinate Geometry Distance – Pythagoras (2/3) Distance from (2,2) to (7,6) = c 2 = a 2 + b 2 c 2 = c 2 = c 2 = 41 c = 41

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Co-ordinate Geometry Distance – Pythagoras (3/3) Distance from (-1,-2) to (5,5) = c 2 = a 2 + b 2 c 2 = c 2 = c 2 = 85 c = 85

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Co-ordinate Geometry Distance – Pythagoras (1/1) Distance from (1,-2) to (7,7) 6 9 c 2 = a 2 + b 2 c 2 = c 2 = c 2 = 117 c = c ≈ 10.8 ≈ 10.8

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Co-ordinate Geometry 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 +( ) 2 (x 1,y 1 ) (x 2,y 2 ) = = = 65 (as surd) Exact ≈ 8.06 (approx) GeoGebra

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Co-ordinate Geometry 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) = = 73 (as surd) Exact

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Co-ordinate Geometry 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 ) = = = 10 (2 decimal places) Approximate = 3.16

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Co-ordinate Geometry 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 ), = ( ), = (5, 2) GeoGebra

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Co-ordinate Geometry 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 ), = ( ), = (8, 3.5)

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

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

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Co-ordinate Geometry 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 Gradient Formula (4/5) 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 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 = = 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 = = -6 2 = -3

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Co-ordinate Geometry 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 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 y -3y 0 = x -3y + 9 x -3y + 9 = 0

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Co-ordinate Geometry 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 c) y - 2x = 0 +2x+2x y = 2x d) 3y + x -1 = 0 -x -x y = -x +1 y = x ÷3 ÷3 ÷3

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

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Co-ordinate Geometry 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 y = -4x - 1 y = 12x + 7

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Co-ordinate Geometry 26 y = x Linear Equations (5/6) Write the equation as y=mx+b and state m and b. a) y – 2x = 1 b) 2y = 3x x+2x y = 2x + 1 m = 2 b = 1 ÷2÷2÷2 72 m = b = 32 72

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Co-ordinate Geometry 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 = 2x = 5 Yes 24 = x4 = 3x = 10 No 3 - 2x1 = 1 1 = 1 Yes 5 = 5x = 6 No

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Co-ordinate Geometry 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 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 y - 3 = -5x y = -5x + 3 m 1 = -5 m 2 = -5 They are parallel !

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Co-ordinate Geometry 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 y - 3 = -3x y = -3x + 3 m 1 = -2 m 2 = -3 They are NOT parallel !

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Co-ordinate Geometry 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 m1 = 2m1 = 2m1 = 2m1 = 2 m 2 = -2 They are NOT parallel ! 4y = -8x –

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Co-ordinate Geometry 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 m 1 = -2 m 2 = -2 They are parallel ! 4y = -8x –

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