# Ratio Formula and Mid Point Formula

Let ${\text{P}}\left( {{{\text{x}}_1},{{\text{y}}_1}} \right)$ and ${\text{Q}}\left( {{{\text{x}}_2},{{\text{y}}_2}} \right)$ be any two points on the line.

Let a point ${\text{R}}\left( {{\text{x}},{\text{y}}} \right)$ be the point which divides PQ in the ratio ${{\text{K}}_1}:{{\text{K}}_2}$ i.e. PR : PQ = ${{\text{K}}_1}:{{\text{K}}_2}$

From P, R and Q draw PM, RN and OL perpendicular to the X-Axis.

From ‘P’ draw ${\text{PS}} \bot$to RN.

From ‘R’ draw ${\text{RT}} \bot$to QL.

Since the right triangles$\Delta {\text{PRS}}$ and $\Delta {\text{RQT}}$
$\begin{gathered} m\angle {\text{SRP}} = m\angle {\text{TRQ}} \\ m\angle {\text{PSR}} = m\angle {\text{RTQ}} \\ m\angle {\text{SRP}} = m\angle {\text{TQR}} \\ \end{gathered}$

$\therefore$ $\Delta {\text{PRS}}$ and $\Delta {\text{RQT}}$ are similar triangles.

$\begin{gathered} {\text{PS}}:{\text{RT}} = {\text{RS}}:{\text{QT}} = {\text{PR}}:{\text{RQ}} \\ \frac{{{\text{PS}}}}{{{\text{RT}}}} = \frac{{{\text{RS}}}}{{{\text{QT}}}} = \frac{{{\text{PR}}}}{{{\text{RQ}}}} \\ \frac{{{\text{MN}}}}{{{\text{NL}}}} = \frac{{{\text{RN}} – {\text{SN}}}}{{{\text{QL}} – {\text{TL}}}} = \frac{{{{\text{K}}_{\text{1}}}}}{{{{\text{K}}_{\text{2}}}}} \\ \frac{{{\text{ON}} – {\text{OM}}}}{{{\text{OL}} – {\text{ON}}}} = \frac{{{\text{RN}} – {\text{SN}}}}{{{\text{QL}} – {\text{TL}}}} = \frac{{{{\text{K}}_{\text{1}}}}}{{{{\text{K}}_{\text{2}}}}} \\ \frac{{{\text{x}} – {{\text{x}}_{\text{1}}}}}{{{{\text{x}}_{\text{2}}} – {\text{x}}}} = \frac{{{\text{y}} – {{\text{y}}_{\text{1}}}}}{{{{\text{y}}_{\text{2}}} – {\text{y}}}} = \frac{{{{\text{K}}_{\text{1}}}}}{{{{\text{K}}_{\text{2}}}}} \\ \end{gathered}$

$\frac{{{\text{x}} – {{\text{x}}_{\text{1}}}}}{{{{\text{x}}_{\text{2}}} – {\text{x}}}} = \frac{{{{\text{K}}_{\text{1}}}}}{{{{\text{K}}_{\text{2}}}}}$

Cross Multiplication
$\begin{gathered} {{\text{K}}_{\text{2}}}\left( {{\text{x}} – {{\text{x}}_{\text{1}}}} \right) = {{\text{K}}_{\text{1}}}\left( {{{\text{x}}_{\text{2}}} – {\text{x}}} \right) \\ {{\text{K}}_{\text{2}}}{\text{x}} – {{\text{K}}_{\text{2}}}{{\text{x}}_{\text{1}}} = {{\text{K}}_{\text{1}}}{{\text{x}}_{\text{2}}} – {{\text{K}}_{\text{1}}}{\text{x}} \\ {{\text{K}}_{\text{1}}}{\text{x}} + {{\text{K}}_{\text{2}}}{\text{x}} = {{\text{K}}_{\text{1}}}{{\text{x}}_{\text{2}}} + {{\text{K}}_{\text{2}}}{{\text{x}}_{\text{1}}} \\ \left( {{{\text{K}}_{\text{1}}} + {{\text{K}}_{\text{2}}}} \right){\text{x}} = {{\text{K}}_{\text{1}}}{{\text{x}}_{\text{2}}} + {{\text{K}}_{\text{2}}}{{\text{x}}_{\text{1}}} \\ {\text{x}} = \frac{{{{\text{K}}_{\text{1}}}{{\text{x}}_{\text{2}}} + {{\text{K}}_{\text{2}}}{{\text{x}}_{\text{1}}}}}{{{{\text{K}}_{\text{1}}} + {{\text{K}}_{\text{2}}}}} \\ \end{gathered}$

$\frac{{{\text{y}} – {{\text{y}}_{\text{1}}}}}{{{{\text{y}}_{\text{2}}} – {\text{y}}}} = \frac{{{{\text{K}}_{\text{1}}}}}{{{{\text{K}}_{\text{2}}}}}$

Cross Multiplication
$\begin{gathered} {{\text{K}}_{\text{2}}}\left( {{\text{y}} – {{\text{y}}_{\text{1}}}} \right) = {{\text{K}}_{\text{1}}}\left( {{{\text{y}}_{\text{2}}} – {\text{y}}} \right) \\ {{\text{K}}_{\text{2}}}{\text{y}} – {{\text{K}}_{\text{2}}}{{\text{y}}_{\text{1}}} = {{\text{K}}_{\text{1}}}{{\text{y}}_{\text{2}}} – {{\text{K}}_{\text{1}}}{\text{y}} \\ {{\text{K}}_{\text{1}}}{\text{y}} + {{\text{K}}_{\text{2}}}{\text{y}} = {{\text{K}}_{\text{1}}}{{\text{y}}_{\text{2}}} + {{\text{K}}_{\text{2}}}{{\text{y}}_{\text{1}}} \\ \left( {{{\text{K}}_{\text{1}}} + {{\text{K}}_{\text{2}}}} \right){\text{y}} = {{\text{K}}_{\text{1}}}{{\text{y}}_{\text{2}}} + {{\text{K}}_{\text{2}}}{{\text{y}}_{\text{1}}} \\ {\text{y}} = \frac{{{{\text{K}}_{\text{1}}}{{\text{y}}_{\text{2}}} + {{\text{K}}_{\text{2}}}{{\text{y}}_{\text{1}}}}}{{{{\text{K}}_{\text{1}}} + {{\text{K}}_{\text{2}}}}} \\ \end{gathered}$

Required Point
$\boxed{{\text{R}}\left( {{\text{x}},{\text{y}}} \right) = \left( {\frac{{{{\text{K}}_{\text{1}}}{{\text{x}}_{\text{2}}} + {{\text{K}}_{\text{2}}}{{\text{x}}_{\text{1}}}}}{{{{\text{K}}_{\text{1}}} + {{\text{K}}_{\text{2}}}}},\frac{{{{\text{K}}_{\text{1}}}{{\text{y}}_{\text{2}}} + {{\text{K}}_{\text{2}}}{{\text{y}}_{\text{1}}}}}{{{{\text{K}}_{\text{1}}} + {{\text{K}}_{\text{2}}}}}} \right)}$