Coordinate geometry is a mathematical technique that allows any linear function to be represented by a pair of numbers or coordinates. It uses algebraic methods to solve geometrical problems.

Every linear function or equation can be represented by y = mx + c where m and c are numerical values for a straight line.

## Distance Formula

In a number plane, if a point A is represented by (x_{1}, y_{1}) then x_{1} and y_{1} are called the coordinates the point A. Now, let us consider point A (x_{1}, y_{1}) and point B (x_{2}, y_{2}). We can find the distance between the points A and B using Pythagoras rule which states that

AB^{2} =

This is called distance formula.

The distance AB between A (x_{1}, y_{1}) and B (x_{2}, y_{2}) is give by

Now let us look at a few examples.

**Example 1****:** Find the distance between (3, 5) and (4, 8).

Here x_{1 }= 3, y_{1 }= 5, x_{2 }= 4, y_{2 }= 8. Applying the distance formula –

= units

**Example 2****:** Which is closer to (4, 0) – the point A (1, 3) or B (-2, 2)?

With A (1, 3), x_{1 }= 1, y_{1 }= 3 and with B (-2, 2), x_{2 }= 4, y_{2 }= 0

Distance of given point (4, 0) from A is

=

= units

= 4.2426 units

Distance of given point (4,0) from B is x_{1 }= -2, y_{2 }= 2

=

=

=

=

= 6.3245 units

Since d_{1} is lesser than d_{2}, the point A is closer to (4, 0) than point B.

**Example 3****:** Prove that the triangle ABC is an isosceles triangle

Given A (2, 0), B (-2, 0), and C (-0, +2).

A(2, 0) : x_{1}– = 2, y_{1 }= 0

B(-2, 0) : x_{2}– = -2, y_{2 }= 0

AB =

= = 4

Similarly we can find distances AC and BC.

To get AC –

A(2, 0) : x_{1}– = 2, y_{1 }= 0

C(-0, +2) : x_{2}– = -0, y_{2 }= +2

AC =

= units

To get BC –

B(-2, 0) : x_{1}– = -2, y_{1 }= 0

C(-0, +2) : x_{2}– = -0, y_{2 }= +2

BC =

= units

In an isosceles triangle, two sides are of equal length. In triangle ABC, sides AC and BC have equal length ( units). Hence triangle ABC is an isosceles triangle.

**Example 4****:** A is the point (-2, 7) and B (5, -2). M is halfway between A and B. How far is M from A?

x coordinate of M is halfway between x co-ordinates of A and B

and y coordinate of M is halfway between y co-ordinates of A and B

Therefore the coordinates of the midpoint is

Distance of from A (-2, 7) is

d =

=

=

=

= units