Co-ordinates, Points and Lines

Co-ordinates, Points and Lines

1. Distance between two points

   • Pythagoras Therom
     • a^2 + b^2 = c^2
   • Slightly different formula
     • (x2-x1)^2 + (y2-y1)^2 = distance^2
     • Eg: Distance between (6,4) and (9,8)
       • (9-6)^2 + (8-4)^2
         • 3^2 + 4^2
           • 9+16=25
             • Distance = 5
   • Can leave in surd form
     • Have to if letters involved

2. Mid-points

   • If there is a line with P,Q and M marked on it
     • Then PM=QM
     • Imagine a triangle in which P and Q are corners and R is at a right angle to Q
       • QR is halved, the point called A and a line traced to PQ
       • The same is done with PR giving two smaller triangles in our imaginary triangle
         • Find the Co-ordinate of A and B and you can find M
   • (((x1+x2)/2),((y1+y2)/2))
     • Eg: Find the co-ordinate of the midpoint between (2,11) and (6,15)
       • (((2+6)/2),(11+15)/2))
         • ((8/2),(26/2))
           • (4,13)

3. Gradient

   • Rise/ Run
     • y/x
   • y2-y1/x2-x1
   • Eg Find the gradient of the line joining (3,8) and (5,12)
     • 12-8/5-3
       • 4/2
         • 2

4. Equation of a line

   • Through a given point with Gradient
     • y-y1=m(x-x1) m is the gradient
       • Find the gradient
         • Use point co-ordinates
     • Eg Find the equation of the line connecting (-3,10) and (1-4)
   • Joining two points
   • Various forms
     • y=mx+c
       • m= gradient c= y-intercept
     • ax+by+c=0
       • a,b and c are constants
     • y-y1=m(x-x1)

5. Points of Intersection

6. Parallel or Perpendicular

   • Parallel lines have the same gradient
   • Whereas Perpendicular lines have the negative reciprocal of a gradient
     • Negative reciprocal means changing an integer into a negative fraction and a fraction top heavy
       • 5= -1/5
         • 2/3= -3/2
           • -5=1/5
             • -2/3=3/2

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