Coordinate Geometry

The Distance Between two Points

The length of the line joining the points (x₁, y₁) and (x₂, y₂) is:

Example

Find the distance between the points (5, 3) and (1, 4).
(So in this case, x₂ = 1, x₁ = 5, y₂ = 4 and y₁ = 3).

Distance = Ö[(1 - 5)² + (4 - 3)²]
= Ö[16 + 1] = Ö17

The Midpoint of a Line Joining Two Points

The midpoint of the line joining the points (x₁, y₁) and (x₂, y₂) is:

• [½(x₁ + x₂), ½(y₁ + y₂)]

Example

Find the coordinates of the midpoint of the line joining (1, 2) and (3, 1).

Midpoint = [½(3 + 1), ½(2 + 1)] = (2, 1.5)

The Gradient of a Line Joining Two Points

The gradient of the line joining the points (x₁, y₁) and (x₂, y₂) is:
y₂ - y₁
x₂ - x₁

Example

Find the gradient of the line joining the points (5, 3) and (1, 4).
Gradient = 4 - 3 =  1 =  -0.25
               1 - 5    -4

Parallel and Perpendicular Lines

If two lines are parallel, then they have the same gradient.

If two lines are perpendicular, then the product of the gradients of the two lines is -1.

Example

a) y = 2x + 1
b) y = -½ x + 2
c) ½y = x - 3

The gradients of the lines are 2, -½ and 2 respectively. Therefore (a) and (b) and perpendicular, (b) and (c) are perpendicular and (a) and (c) are parallel.

The Equation of a Line Using One Point and the Gradient

The equation of a line which has gradient m and which passes through the point (x₁, y₁) is:

• y - y₁ = m(x - x₁)

Example

Find the equation of the line with gradient 2 passing through (1, 4).

y - 4 = 2(x - 1)
y - 4 = 2x - 2
y = 2x + 2

Since m = y₂ - y₁
              x₂ - x₁

The equation of a line passing through (x₁, y₁) and (x₂, y₂) can be written as:
y - y₁ = y₂ - y₁
x - x₁    x₂ - x₁

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