Do you know how to find **distance between two points** in coordinate geometry? This is the first formula we are going to learn in coordinate geometry.

Hope that you know how to represent a point in coordinate system. Suppose if we are given any two points and we are asked to find distance between them, then how to find it. Now let us learn **formula** for finding **distance between two points**.

**Formula for finding distance between two points:**

Consider a coordinate system. Now let us take two points in the coordinate system as shown in figure. Let the two points coordinates be P \left( { x }_{ 1 },{ y }_{ 1 } \right) and Q \left( { x }_{ 2 },{ y }_{ 2 } \right) . Now PQ we will get a line segment PQ. Now let us start deriving the **formula** for **distance between two points** i.e. P and Q. for this we have to do a small construction.

Construction:

First draw a perpendicular line from point P to *x*-axis and name the intersecting point as C.

Similarly from point Q draw a perpendicular line to *x*-axis and name the intersecting point as D.

Now through the point P, draw a perpendicular line to the line QD and name the intersecting point as R.

From the diagram,

OD = { x }_{ 2 } units

OC = { x }_{ 1 } units

CD = OD – OC = { x }_{ 2 } – { x }_{ 1 } _{ }units

Therefore, PR = CD = { x }_{ 2 } – { x }_{ 1 } _{ }units

QD = { y }_{ 2 } units

RD = { y }_{ 1 } units

QR = QD – RD= { y }_{ 2 } – { y }_{ 1 } units.

The obtained PQR is a right triangle. So that we can make use of Pythagoras theorem.

According to Pythagoras theorem, we can say that

{ \left( Hyp \right) }^{ 2 } = { \left( side1 \right) }^{ 2 } + { \left( side2 \right) }^{ 2 }

{ \left( PQ \right) }^{ 2 } = { \left( PR \right) }^{ 2 } + { \left( QR \right) }^{ 2 }

{ \left( PQ \right) }^{ 2 } = { \left( { x }_{ 2 }-{ x }_{ 1 } \right) }^{ 2 } + { \left( { y }_{ 2 }-{ y }_{ 1 } \right) }^{ 2 }

PQ = \sqrt { { \left( { x }_{ 2 }-{ x }_{ 1 } \right) }^{ 2 }+{ \left( { y }_{ 2 }-{ y }_{ 1 } \right) }^{ 2 } }

So, finally the **distance between the two points** P and Q is given by the formula PQ = \sqrt { { \left( { x }_{ 2 }-{ x }_{ 1 } \right) }^{ 2 }+{ \left( { y }_{ 2 }-{ y }_{ 1 } \right) }^{ 2 } }

I explained only the basic formula but in this still we have so many cases, that we will see later.

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