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).

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).

[½(4), ½(3)] = (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

**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) is therefore:

__y - y1__ = __y2 - y1__

x - x1 x2 - x1

Coordinate geometry