Distance Between Two Points

In IB Mathematics, the length of the line joining the points (x_{1}, y_{1}) and (x_{2}, y_{2}) is:

**Example**

Find the distance between the points (5, 3) and (1, 4).

(So in this case, x_{2} = 1, x_{1} = 5, y_{2} = 4 and y_{1} = 3).

Distance = √ (1-5)^{2 }+ (4-3)^{2} = √17

The Midpoint of a Line Joining Two Points

The midpoint of the line joining the points (x_{1}, y_{1}) and (x_{2}, y_{2}) is:

**(x _{1}+x_{2})/2 + (y_{1}+_{y2})/2**

**Example**

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

Midpoint = (1+3)/2 + (2+1)/2= (2, 1.5)

The Gradient of a Line Joining Two Points

In IB Mathematics, the gradient of the line joining the points (x_{1}, y_{1}) and (x_{2}, y_{2}) is:

**y _{2} - y_{1}/_{ }x_{2} - x**

_{1}

**Example**

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

Gradient = (4-3)/ (1-5) = -0.25

Parallel and Perpendicular Lines

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

In IB Mathematics, if two lines are **perpendicular**, then the gradients of the two lines are **reciprocals** of each other.

**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

In IB Mathematics, the equation of a line which has gradient m and which passes through the point (x_{1}, y_{1}) is:

**y - y _{1} = m(x- x_{1})**

**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 = y_{2} - y_{1}/x_{2} - x_{1}

In IB Mathematics, the equation of a line pass through(x_{1}, y_{1}) and (x_{2}, y_{2}) can be written as:

This is the end of this topic.