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.


ratio-formula

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 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 triangle.

\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)}

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