Section Formula (Internal Division)

The Section Formula (Internal Division) is a fundamental concept in coordinate geometry. It helps us find the coordinates of a point that divides a line segment internally in a given ratio. Internal division means that the point lies *between* the endpoints of the line segment.

Essentially, it allows us to pinpoint a specific location on a line segment based on a proportional division.

Formulae

Let’s say we have a line segment with endpoints $A(x_1, y_1)$ and $B(x_2, y_2)$. A point $P(x, y)$ divides this line segment internally in the ratio $m:n$. Then the coordinates of point P are given by the following formulas:

• x-coordinate: $x = \frac{mx_2 + nx_1}{m + n}$
• y-coordinate: $y = \frac{my_2 + ny_1}{m + n}$

In short, the section formula is: $(\frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n})$

Examples

Example-1: Find the coordinates of the point that divides the line segment joining the points A(2, 3) and B(5, 7) internally in the ratio 2:1.

Solution:

Here, $x_1 = 2$, $y_1 = 3$, $x_2 = 5$, $y_2 = 7$, $m = 2$, and $n = 1$.

Using the section formula:

• $x = \frac{(2)(5) + (1)(2)}{2 + 1} = \frac{10 + 2}{3} = \frac{12}{3} = 4$
• $y = \frac{(2)(7) + (1)(3)}{2 + 1} = \frac{14 + 3}{3} = \frac{17}{3}$

Therefore, the coordinates of the point are $(4, \frac{17}{3})$.

Example-2: Determine the coordinates of the point that divides the line segment joining the points C(-1, 4) and D(3, -2) internally in the ratio 3:2.

Solution:

Here, $x_1 = -1$, $y_1 = 4$, $x_2 = 3$, $y_2 = -2$, $m = 3$, and $n = 2$.

Using the section formula:

• $x = \frac{(3)(3) + (2)(-1)}{3 + 2} = \frac{9 – 2}{5} = \frac{7}{5}$
• $y = \frac{(3)(-2) + (2)(4)}{3 + 2} = \frac{-6 + 8}{5} = \frac{2}{5}$

Therefore, the coordinates of the point are $(\frac{7}{5}, \frac{2}{5})$.

Common mistakes by students

• Reversing the ratio: Incorrectly assigning the values of $m$ and $n$ to the coordinates. Make sure $m$ is associated with the second point’s coordinates and $n$ with the first point’s.
• Miscalculation: Simple arithmetic errors in the numerator or denominator. Always double-check your calculations, especially with negative numbers.
• Not understanding internal division: Confusing internal division with external division. Ensure you’re using the correct formula for the given problem.

Real Life Application

The section formula has various real-world applications:

• Computer Graphics: In computer graphics, this formula is used to generate smooth lines and curves, and to divide objects into proportional segments for rendering.
• Mapping and Navigation: For determining locations or dividing distances on a map or in navigation systems. For example, finding a landmark between two points.
• Construction and Engineering: Calculating the position of a support beam or dividing a line into segments.
• Art and Design: To create visually balanced compositions, artists and designers may employ the section formula to place elements proportionally.

Fun Fact

The section formula is directly related to the concept of weighted averages. Think of the ratio $m:n$ as the “weights” applied to the coordinates of the endpoints. The formula then calculates a weighted average to find the position of the dividing point.

Recommended YouTube Videos for Deeper Understanding

Q.1 If point $P$ divides the line segment joining $A(2, 3)$ and $B(7, 8)$ internally in the ratio $2:3$, what are the coordinates of $P$?

Check Solution

Ans: A

Using the section formula: $P = \left(\frac{2 \cdot 7 + 3 \cdot 2}{2+3}, \frac{2 \cdot 8 + 3 \cdot 3}{2+3}\right) = \left(\frac{20}{5}, \frac{25}{5}\right) = (4, 5)$

Q.2 Find the coordinates of the point that divides the line segment joining the points $C(-1, 4)$ and $D(5, -2)$ internally in the ratio $1:2$.

Check Solution

Ans: B

Using the section formula: $P = \left(\frac{1 \cdot 5 + 2 \cdot (-1)}{1+2}, \frac{1 \cdot (-2) + 2 \cdot 4}{1+2}\right) = \left(\frac{3}{3}, \frac{6}{3}\right) = (1, 2)$

Q.3 Given that the point $E(x, y)$ divides the line segment joining $F(3, -4)$ and $G(-6, 2)$ internally in the ratio $1:3$, find the value of $x+y$.

Check Solution

Ans: A

Using the section formula: $E = \left(\frac{1 \cdot (-6) + 3 \cdot 3}{1+3}, \frac{1 \cdot 2 + 3 \cdot (-4)}{1+3}\right) = \left(\frac{3}{4}, \frac{-10}{4}\right)$. Thus, $x+y = \frac{3}{4} – \frac{10}{4} = -\frac{7}{4}$

Q.4 The point $H$ divides the line segment joining $I(-2, 5)$ and $J(4, -1)$ in the ratio $k:1$ internally. If the x-coordinate of $H$ is $1$, what is the value of $k$?

Check Solution

Ans: B

Using the section formula, $1 = \frac{k \cdot 4 + 1 \cdot (-2)}{k+1}$. Then $k+1 = 4k-2$, and $3k=3$, so $k=1$.

Q.5 If the point $K$ divides the line segment $L(a, b)$ and $M(c, d)$ internally in the ratio $p:q$, and $K$ has coordinates $(x, y)$, which of the following is true?

Check Solution

Ans: A

Using the section formula, $K = \left(\frac{pc + qa}{p+q}, \frac{pd + qb}{p+q}\right)$.

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