# Section Formula

### Section Formula:

#### Definition:

The Section Formula is used to find the coordinates of a point that divides a line segment joining two given points in a specific ratio.

#### Formula:

Let $P\left(x,y\right)$ be the point dividing the line segment joining $A\left({x}_{1},{y}_{1}\right)$ and $B\left({x}_{2},{y}_{2}\right)$ in the ratio $m:n$. The coordinates of $P$ are given by:

$x=\frac{m{x}_{2}+n{x}_{1}}{m+n}$ $y=\frac{m{y}_{2}+n{y}_{1}}{m+n}$

#### Example:

Consider two points A(3, 5) and B(7, 9). Find the coordinates of a point P that divides AB in the ratio 2:3.

#### Steps to Solve:

1. Identify Coordinates:

• Point A: ${x}_{1}=3,{y}_{1}=5$
• Point B: ${x}_{2}=7,{y}_{2}=9$
• Ratio: $m=2,n=3$
2. Apply Section Formula: $x=\frac{m{x}_{2}+n{x}_{1}}{m+n}$ $y=\frac{m{y}_{2}+n{y}_{1}}{m+n}$

3. Substitute Values: $x=\frac{2×7+3×3}{2+3}$ $y=\frac{2×9+3×5}{2+3}$

4. Perform Arithmetic: $x=\frac{14+9}{5}=\frac{23}{5}=4.6$ $y=\frac{18+15}{5}=\frac{33}{5}=6.6$

5. Final Answer: The coordinates of point P, which divides the line segment AB in the ratio 2:3, are approximately (4.6, 6.6).

### Interpretation:

Using the Section Formula, we've found the coordinates of point P that divides the line segment AB in the ratio 2:3. These coordinates represent the location of point P along the line segment joining points A and B in the given ratio.