Coordinate Geometry

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Midpoints

Point A is (-1,2)   Point B is (3,-4)   M is the midpoint of the line joining A and B.

  • The x-coordinate of M is (-1 + 3) divided by 2 = 1
  • The y-coordinate of M is (2 + -4) divided by 2 = -1            The midpoint is (1,-1)

A = (x1,y1)  B = (x2,y2)   The midpoint of AB is ((x1 + x2) divided by 2, (y1 + y2) divided by 2)

Point A is (-5,3)   Point B is (4,0)   Find the length of AB  

  • The x-axis = 9   The y-axis = 3 9 squared = 81   3 squared = 9   81 + 9 = 90   root 90 = 9.5   Line AB = 9.5

A = (x1,y1)  B = (x2,y2)   The midpoint of AB is root(x2 - x1) squared + (y2 - y1) squared

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Gradient

Gradient = up divided by across   The gradient of a line with points (-2,0) and (0,2), 2 divided by 2 = 1   This is a 45 degree line

A = (x1,y1) B = (x2,y2) The gradient of AB is y1 - y2 divided by x1 -xor y2 - y1 divided by x2 - x1

dy/dx means the gradient of a line

Perpendicular is the same as normal   The normal of dy/dx 2 is -1/2

The normal of dy/dx x/y is -y/x   x = 2   y = 1

Two lines with equal gradients are parallel

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y = mx + c

m is the gradient   y = 2x + c has a gradient of 2

c is the y-intercept   y = mx + 4 crosses the y axis at (0,4)

First rearrange the equation to write y = mx + c   make sure the y is an integer (whole number)

If x is a fraction the denominator is the across value while the numerator is the up value (n/d)

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Equation of a Line

If we know one point on a line and the gradient the formula y - y1 = m(x - x1) is used to find the equation or the y-intercept

The gradient is 2 and the  point is (1,3)   y - 3 = 2(x -1)   y - 3 = 2x - 2   y = 2x + 1

If we don't know the gradient we can still use the formula with two points as well

The points are (2,4) and (7,3)   m = (3-4)/(7-2)   m = -1/5   y - 4 = -1/5(x - 2)   5y - 20 = -x + 2   5y = -x + 22   it can be writen x + 5y = 22

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Intersecting Lines

Unless two lines are parallel they will intersect

First make both lines equal each other through y or x, then find the value of x or y

x + y = 6  2x - y = 9    y = -x + 6  y = 2x - 9    2x - 9 = -x + 6   3x = 15   x = 5

Then substitute the found number into either equation

5 + y = 6   y = 1   the point where the lines intersect is (5,1)

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