## Divisor of a Line Segment

 7.1 Divisor of a Line Segment

Ratio Division
 Figure

 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

 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:

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

## Divisor of a Line Segment

 7.1 Divisor of a Line Segment

Ratio Division
 Figure

 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

 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:

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