Coordinate Geometry Section Formula

What is Coordinate Geometry?

Coordinate Geometry, or analytic geometry, connects algebra and geometry by representing geometric shapes and problems using equations. Each point on a plane is represented by an ordered pair:

$\boldsymbol{(x, y)}$

where:

$\boldsymbol{x}$ is the horizontal distance (abscissa).
$\boldsymbol{y}$ is the vertical distance (ordinate).

Why Learn the Section Formula?

The section formula helps locate a point that divides a line segment joining two points in a given ratio. It is widely applicable in:

Engineering and Design: To proportionally divide structures or objects.
Navigation and Mapping: Locating intermediate points between destinations.
Graphics and Animation: Rendering positions in 2D models.
Physics: Analyzing motion along a straight line.

Section Formula

The section formula provides the coordinates of a point $\boldsymbol{P(x, y)}$ dividing a line segment joining two points $\boldsymbol{(x_1, y_1)}$ and $\boldsymbol{(x_2, y_2)}$ in the ratio $\boldsymbol{m:n}$ :

$\displaystyle\boldsymbol{P(x, y) = \left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right)}$

Special Cases:

Midpoint Formula:
When the dividing point $\boldsymbol{P(x, y)}$ divides the line segment into equal parts ( $\boldsymbol{m = n}$ ): $\displaystyle\boldsymbol{M(x, y) = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)}$
Centroid of a Triangle:
The point $\boldsymbol{G(x, y)}$ that divides each median of a triangle into a ratio of $\boldsymbol{2:1}$ (vertex to midpoint): $\displaystyle\boldsymbol{G(x, y) = \left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right)}$
Distance Formula:
The distance $\boldsymbol{d}$ between two points $\boldsymbol{(x_1, y_1)}$ and $\boldsymbol{(x_2, y_2)}$ is given by: $\boldsymbol{d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}}$
External Division Formula:
When a point $\boldsymbol{P(x, y)}$ divides the line segment externally in the ratio $\boldsymbol{m:n}$ , the formula is: $\displaystyle\boldsymbol{P(x, y) = \left(\frac{mx_2 - nx_1}{m-n}, \frac{my_2 - ny_1}{m-n}\right)}$
Slope of a Line:
The slope $\boldsymbol{m}$ of a line passing through two points $\boldsymbol{(x_1, y_1)}$ and $\boldsymbol{(x_2, y_2)}$ is: $\displaystyle\boldsymbol{m = \frac{y_2 - y_1}{x_2 - x_1}}$

Section Formula

Definition:

The section formula helps determine the coordinates of a point $\boldsymbol{P(x, y)}$ that divides a line segment joining two points $\boldsymbol{(x_1, y_1)}$ and $\boldsymbol{(x_2, y_2)}$ in a given ratio $\boldsymbol{m:n}$ . This formula is crucial for solving problems in geometry where proportional division of line segments is required.

Formula:

The coordinates of the point $\boldsymbol{P(x, y)}$ are given by:

$\displaystyle\boldsymbol{P(x, y) = \left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right)}$

Where:

$\boldsymbol{P(x, y)}$ : Represents the coordinates of the dividing point.
$\boldsymbol{(x_1, y_1)}$ and $\boldsymbol{(x_2, y_2)}$ : Are the coordinates of the endpoints of the line segment.
$\boldsymbol{m:n}$ : Denotes the ratio in which the point divides the line segment.
$\boldsymbol{m}$ and $\boldsymbol{n}$ are positive constants.

Derivation of the Section Formula:

To derive the formula for the coordinates of a point $\boldsymbol{P(x, y)}$ that divides a line segment joining two points $\boldsymbol{A(x_1, y_1)}$ and $\boldsymbol{B(x_2, y_2)}$ in the ratio $\boldsymbol{m:n}$ .

Step-by-Step Derivation

Consider a Line Segment:
- Let $\boldsymbol{A(x_1, y_1)}$ and $\boldsymbol{B(x_2, y_2)}$ be the endpoints of the line segment.
- Let $\boldsymbol{P(x, y)}$ be the point dividing $\boldsymbol{AB}$ in the ratio $\boldsymbol{m:n}$ .
Proportionality:
- The division ratio implies: $\displaystyle\boldsymbol{\frac{\text{AP}}{\text{PB}} = \frac{m}{n}}$
- This means that the distance $\boldsymbol{\text{AP}}$ is $\boldsymbol{m}$ parts, and $\boldsymbol{\text{PB}}$ is $\boldsymbol{n}$ parts.
Use of Coordinates:
- In coordinate geometry, the distances along the x-axis and y-axis are proportional to the division ratio.
- Let the x-coordinate of $\boldsymbol{P}$ be $\boldsymbol{x}$ , and the y-coordinate be $\boldsymbol{y}$ .
Apply Proportionality Along the X-Axis:
- The position of $\boldsymbol{P}$ along the x-axis is determined by the weighted average of $\boldsymbol{x_1}$ and $\boldsymbol{x_2}$ : $\displaystyle\boldsymbol{\frac{x - x_1}{x_2 - x} = \frac{m}{n}}$
- Cross-multiply to get: n $\boldsymbol{n(x - x_1) = m(x_2 - x)}$
- Expand and rearrange: $\boldsymbol{nx - nx_1 = mx_2 - mx}$ $\boldsymbol{nx + mx = mx_2 + nx_1}$ $\boldsymbol{x(m + n) = mx_2 + nx_1}$
- Solve for $\boldsymbol{x}$ : $\displaystyle\boldsymbol{x = \frac{mx_2 + nx_1}{m + n}}$
Apply Proportionality Along the Y-Axis:
- Similarly, for the y-coordinate of $\boldsymbol{P}$ : $\displaystyle\boldsymbol{\frac{y - y_1}{y_2 - y} = \frac{m}{n}}$
- Cross-multiply and solve as done for $\boldsymbol{x}$ : $\boldsymbol{n(y - y_1) = m(y_2 - y)}$ $\boldsymbol{ny - ny_1 = my_2 - my}$ $\boldsymbol{ny + my = my_2 + ny_1}$ $\boldsymbol{y(m + n) = my_2 + ny_1}$
- Solve for $\boldsymbol{y}$ : $\displaystyle\boldsymbol{y = \frac{my_2 + ny_1}{m + n}}$
Combine Results:
- The coordinates of $\boldsymbol{P}$ are: $\displaystyle\boldsymbol{P(x, y) = \left(\frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n}\right)}$

Applications of the Section Formula:

Internal Division:
- When the point lies between the two endpoints of the line segment, the section formula is applied as:
  $\displaystyle\boldsymbol{P(x, y) = \left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right)}$
External Division:
- When the point divides the line segment externally, the formula becomes:
  $\displaystyle\boldsymbol{P(x, y) = \left(\frac{mx_2 - nx_1}{m-n}, \frac{my_2 - ny_1}{m-n}\right)}$
  Here, $\boldsymbol{m > n}$ to ensure valid results.
Special Cases:
- Midpoint Formula:
  If $\boldsymbol{m = n}$ , the formula simplifies to:
  $\displaystyle\boldsymbol{M(x, y) = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)}$
- Centroid of a Triangle:
  When dividing the medians of a triangle in the ratio $\boldsymbol{2:1}$ :
  $\displaystyle\boldsymbol{G(x, y) = \left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right)}$
Distance Formula (for verifying calculations):
The distance between two points $\boldsymbol{(x_1, y_1)}$ and $\boldsymbol{(x_2, y_2)}$ is:
$\boldsymbol{d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}}$
Slope of a Line:
The slope of a line passing through two points $\boldsymbol{(x_1, y_1)}$ and $\boldsymbol{(x_2, y_2)}$ is:
$\displaystyle\boldsymbol{m = \frac{y_2 - y_1}{x_2 - x_1}}$

Types of Division

Internal Division

Internal division occurs when the point $\boldsymbol{P(x, y)}$ lies between the two endpoints of a line segment $\boldsymbol{(x_1, y_1)}$ and $\boldsymbol{(x_2, y_2)}$ . This means that the point divides the line segment internally in a specified ratio $\boldsymbol{m:n}$ .

Formula for Internal Division:

$\displaystyle\boldsymbol{P(x, y) = \left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right)}$

Explanation:
- $\boldsymbol{m}$ and $\boldsymbol{n}$ represent the weights of division, such that: $\displaystyle\boldsymbol{\frac{\text{AP}}{\text{PB}} = \frac{m}{n}}$
- The formula calculates the weighted average of the coordinates of $\boldsymbol{(x_1, y_1)}$ and $\boldsymbol{(x_2, y_2)}$ .
Example: Find the coordinates of a point dividing the line segment joining $\boldsymbol{(2, 3)}$ and $\boldsymbol{(8, 7)}$ in the ratio $\boldsymbol{3:2}$ .
- Using the internal division formula: $\displaystyle\boldsymbol{P(x, y) = \left(\frac{3(8) + 2(2)}{3+2}, \frac{3(7) + 2(3)}{3+2}\right) = \left(\frac{24 + 4}{5}, \frac{21 + 6}{5}\right) = \left(\frac{28}{5}, \frac{27}{5}\right)}$
- $\boldsymbol{P(x, y) = \left(5.6, 5.4\right)}$ .

External Division

External division occurs when the point $\boldsymbol{P(x, y)}$ lies outside the line segment, such that it divides the line segment $\boldsymbol{(x_1, y_1)}$ and $\boldsymbol{(x_2, y_2)}$ externally in a specified ratio $\boldsymbol{m:n}$ . This type of division happens when the line segment is extended beyond one of the endpoints.

Formula for External Division:

$\displaystyle\boldsymbol{P(x, y) = \left(\frac{mx_2 - nx_1}{m-n}, \frac{my_2 - ny_1}{m-n}\right)}$

Explanation:
- $\boldsymbol{m}$ and $\boldsymbol{n}$ still represent the weights of division, but the subtraction in the formula reflects the external division.
- For valid results, $\boldsymbol{m > n}$ to ensure $\boldsymbol{m-n > 0}$ .
Example: Find the coordinates of a point dividing the line segment joining $\boldsymbol{(4, 6)}$ and $\boldsymbol{(10, 12)}$ externally in the ratio $\boldsymbol{5:3}$ .
- Using the external division formula: $\displaystyle\boldsymbol{P(x, y) = \left(\frac{5(10) - 3(4)}{5-3}, \frac{5(12) - 3(6)}{5-3}\right) = \left(\frac{50 - 12}{2}, \frac{60 - 18}{2}\right) = \left(\frac{38}{2}, \frac{42}{2}\right)}$
- $\boldsymbol{P(x, y) = \left(19, 21\right)}$ .

Comparison of Internal and External Division

Aspect	Internal Division	External Division
Position of Point	Lies between the endpoints of the line segment.	Lies outside the endpoints, beyond the line segment.
Formula	$\displaystyle\boldsymbol{\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right)}$	$\displaystyle\boldsymbol{\left(\frac{mx_2 - nx_1}{m-n}, \frac{my_2 - ny_1}{m-n}\right)}$
Ratio Condition	$\boldsymbol{m}$ and $\boldsymbol{n}$ are both positive.	$\boldsymbol{m > n}$ for valid results.

Solved Examples

Example 1: Internal Division

Problem:
Find the point dividing a line segment joining $\boldsymbol{(2, 3)}$ and $\boldsymbol{(8, 7)}$ in the ratio $\boldsymbol{2:3}$ .

Solution: Using the section formula for internal division:

$\displaystyle\boldsymbol{P(x, y) = \left(\frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n}\right)}$

Substitute:

$\boldsymbol{(x_1, y_1) = (2, 3)}$
$\boldsymbol{(x_2, y_2) = (8, 7)}$
$\boldsymbol{m = 2}$ , $\boldsymbol{n = 3}$

$\displaystyle\boldsymbol{P(x, y) = \left(\frac{2(8) + 3(2)}{2 + 3}, \frac{2(7) + 3(3)}{2 + 3}\right)}$

Simplify:

$\displaystyle\boldsymbol{P(x, y) = \left(\frac{16 + 6}{5}, \frac{14 + 9}{5}\right) = \left(\frac{22}{5}, \frac{23}{5}\right)}$ $\boldsymbol{P(x, y) = \left(4.4, 4.6\right)}$

Example 2: External Division

Problem:
Determine the coordinates of a point dividing a line segment joining $\boldsymbol{(5, 2)}$ and $\boldsymbol{(11, 8)}$ externally in the ratio $\boldsymbol{3:1}$ .

Solution: Using the section formula for external division:

$\displaystyle\boldsymbol{P(x, y) = \left(\frac{mx_2 - nx_1}{m - n}, \frac{my_2 - ny_1}{m - n}\right)}$

Substitute:

$\boldsymbol{(x_1, y_1) = (5, 2)}$
$\boldsymbol{(x_2, y_2) = (11, 8)}$
$\boldsymbol{m = 3}$ , $\boldsymbol{n = 1}$

$\displaystyle\boldsymbol{P(x, y) = \left(\frac{3(11) - 1(5)}{3 - 1}, \frac{3(8) - 1(2)}{3 - 1}\right)}$

Simplify:

$\displaystyle\boldsymbol{P(x, y) = \left(\frac{33 - 5}{2}, \frac{24 - 2}{2}\right) = \left(\frac{28}{2}, \frac{22}{2}\right)}$ $\boldsymbol{P(x, y) = \left(14, 11\right)}P(x,y)=(14,11) <h4>Example 3: Application to Geometry Problems</h4> <ol> <li style="font-weight: 400;" aria-level="1">Verifying Collinearity: Check if points$ \boldsymbol{A(1, 2)} $,$ \boldsymbol{B(4, 5)} $, and$ \boldsymbol{C(7, 8)} $are collinear. <ul> <li style="font-weight: 400;" aria-level="2">Find the slope between$ \boldsymbol{A} $and$ \boldsymbol{B} $:$ \displaystyle\boldsymbol{m_1 = \frac{y_2 – y_1}{x_2 – x_1} = \frac{5 – 2}{4 – 1} = \frac{3}{3} = 1} $</li> <li style="font-weight: 400;" aria-level="2">Find the slope between$ \boldsymbol{B} $and$ \boldsymbol{C} $:$ \displaystyle\boldsymbol{m_2 = \frac{y_2 – y_1}{x_2 – x_1} = \frac{8 – 5}{7 – 4} = \frac{3}{3} = 1} $</li> <li style="font-weight: 400;" aria-level="2">Since$ \boldsymbol{m_1 = m_2} $, the points are collinear.</li> </ul> </li> <li style="font-weight: 400;" aria-level="1">Finding the Midpoint: Find the midpoint of a line segment joining$ \boldsymbol{(-2, 4)} $and$ \boldsymbol{(6, -8)} $. <ul> <li style="font-weight: 400;" aria-level="2">Using the midpoint formula:$ \displaystyle\boldsymbol{M(x, y) = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)} $</li> <li style="font-weight: 400;" aria-level="2">Substitute:$ \displaystyle\boldsymbol{M(x, y) = \left(\frac{-2 + 6}{2}, \frac{4 – 8}{2}\right) = \left(\frac{4}{2}, \frac{-4}{2}\right) = \left(2, -2\right)}$