Division of a Line Segment

 

7.1

Division of a Line Segment

 
 

 

A point \(P(x,y)\) that divides a line segment joining 

\(A(x_1,y_1)\) and

\(B(x_2,y_2)\) in the ratio of

\(m:n\) is given by

 

\(\boxed{P=\left(\dfrac{nx_1+mx_2}{m+n},\dfrac{ny_1+my_2}{m+n} \right)}\)

 
 

 
For the case \(m=n\), the point is a midpoint and is given by
 

\(\boxed{M=\left(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2} \right)}\)

 

Example:

Find the coordinates of point \(P\) which divides the straight line \(AB\) in the ratio \(AP:PB\).

 

\(\begin{aligned} &A(-3,2),B(7,17);\\\\&AP:PB=2:3 \end{aligned}\)

 

\(\begin{aligned} P&=\left(\dfrac{2(7)+3(-3)}{2+3},\dfrac{2(17)+3(2)}{2+3} \right) \\\\ &= (1,8). \end{aligned}\)

 

Division of a Line Segment

 

7.1

Division of a Line Segment

 
 

 

A point \(P(x,y)\) that divides a line segment joining 

\(A(x_1,y_1)\) and

\(B(x_2,y_2)\) in the ratio of

\(m:n\) is given by

 

\(\boxed{P=\left(\dfrac{nx_1+mx_2}{m+n},\dfrac{ny_1+my_2}{m+n} \right)}\)

 
 

 
For the case \(m=n\), the point is a midpoint and is given by
 

\(\boxed{M=\left(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2} \right)}\)

 

Example:

Find the coordinates of point \(P\) which divides the straight line \(AB\) in the ratio \(AP:PB\).

 

\(\begin{aligned} &A(-3,2),B(7,17);\\\\&AP:PB=2:3 \end{aligned}\)

 

\(\begin{aligned} P&=\left(\dfrac{2(7)+3(-3)}{2+3},\dfrac{2(17)+3(2)}{2+3} \right) \\\\ &= (1,8). \end{aligned}\)