Divisor of a Line Segment

7.1

Divisor of a Line Segment

Ratio Division

Figure

A graph featuring two lines, one of which ascends, representing the division of a line segment between two points in a ratio.

Description

The point \(P(x,y)\) divides the line segment joining two points \(A(x_1,y_1)\) and \(B(x_2,y_2)\) in the ratio \(m:n\).
The coordinates of point \(P(x,y)\) are given by:

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

Midpoint (Ratio \(1:1\))

Figure

A graph displaying two lines, with one line ascending, and a midpoint M dividing the segment between points A and B equally.

Description

The midpoint \(M(x,y)\) of the line segment joining \(A(x_1,y_1)\) and \(B(x_2,y_2)\) is:

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

Example Problems

Finding a Dividing Point

Given points \(A(2,3)\) and \(B(8,7)\), find the point \(P\) that divides \(AB\) in the ratio \(2:3\).
Solution:

\(\begin{aligned} x&=\dfrac{2\times8+3\times2}{2+3}=\dfrac{16+6}{5}=4.4, \\ \\ y&=\dfrac{2\times7+3\times3}{2+3}=\dfrac{14+9}{5}=4.6. \end{aligned}\)

Point \(P\) is \((4.4,4.6)\).

Midpoint Calculation

Find the midpoint of the line segment joining \((1,2)\) and \((5,6)\).
Solution:

\(M\left( \dfrac{1+5}{2},\dfrac{2+6}{2} \right)=(3,4)\).