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# Coordinate Geometry

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 (x1, y1) then x1 and y1 are called the coordinates the point A. Now, let us consider point A (x1, y1) and point B (x2, y2). We can find the distance between the points A and B using Pythagoras rule which states that

AB2  This is called distance formula.

The distance AB between A (x1, y1) and B (x2, y2) is give by Now let us look at a few examples.

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

Here x= 3, y= 5, x= 4, y= 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, y= 3 and with B (-2, 2), x= 4, y= 0

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

=  4.2426 units

Distance of given point (4,0) from B is x= -2, y= 2 = = = = = 6.3245 units

Since d1 is lesser than d2, 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) : x1– = 2,  y= 0

B(-2, 0) : x2– = -2,  y= 0

AB = = =  4

Similarly we can find distances AC and BC.

To get AC –

A(2, 0) : x1– = 2,  y= 0

C(-0, +2) : x2– = -0,  y= +2

AC  =  = units

To get BC –

B(-2, 0) : x1– = -2,  y= 0

C(-0, +2) : x2– = -0,  y= +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