# Distance between two points formula explained

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.