#### definition

Whenever a point divides a line segment into two, three, four or any number of sections in a particular ratio then the formula used there is know as Section Formula.

#### notes

1) Section Formula-Section Formula is derived as follows,

Let P be a point (x,y) which divides a line segment AB in the ratio `m_1:m_2`

`"AP"/"PB"= m_1/m_2`

Draw AR, PS and BT ⊥ OT

Also, draw AQ ⊥ PS, PC ⊥ BX

AQ=RS (Since AQSR forms a rectangle opposite sides are equal)

RS= OS-OR

Now, after watching carefully, we can see that OS have travelled `x` units and OR have travelled `x_1` units.

RS= (`x-x_1`)

Therefore, AQ= (`x-x_1`)

Similarly, PC=ST= (`x_2-x`)

and QP= PS-QS= (`y-y_1`)

also, BC= BT-CT= (`y_2-y`)

Clearly, ΔAQP and ΔPCB are similar triangle.

Therefore, `"AP"/"PB"= "AQ"/"PC"= "QP"/"CB"`

`m_1/m_2= (x-x_1)/(x_2-x) =( y-y_1)/(y_2-y)`

`m_1/m_2= (x-x_1)/(x_2-x)` and `m_1/m_2= (y-y_1)/y_2-y`

`m_1x_2-m_1x = m_2x-m_2x_1` and `m_1y_2-m_1y = m_2y-m_2y_1`

`m_1x_2+m_2x_1= m_1x+m_2x ` and `m_1y_2+m_2y_1= m_1y+m_2y`

`m_1x_2+m_2x_1= x(m_1+m_2)` and `m_1y_2+m_2y_1= y(m_1+m_2)`

x= `(m_1x_2+m_2x_1)/(m_1+m_2)` and `y= (m_1y_2+m_2y_1)/(m_1+m_2)`

`P(x,y)= [(m_1x_2+m_2x_1)/(m_1+m_2), (m_1y_2+m_2y_1)/(m_1+m_2)]`

2) Midpoint Formula-

Here let's say P is midpoint and divides AB in equal ratio i.e 1:1

Thus, AP:PB= `m_1:m_2= 1:1`

`P(x,y)= [(1x_2+1x_1)/(1+1), (1y_2+1y_1)/(1+1)]`

`P(x,y)= [(x_2+x_1)/2, (y_2+y_1)/2]`

Therefore, midpoint formula= `[(x_2+x_1)/2, (y_2+y_1)/2]`