103 LearnersLast updated on August 10th, 2025
Section Formula in Mathematics
In geometry, the section formula is used to find the coordinates of a point dividing a line segment in a given ratio. This concept is essential for understanding the division of lines in coordinate geometry. In this topic, we will learn how to apply the section formula.
List of Formulas for the Section Formula
The section formula helps in finding the coordinates of a point dividing a line segment internally or externally. Let’s learn the formulas to calculate these coordinates.
Section Formula for Internal Division
The section formula for internal division finds the coordinates of a point dividing a line segment internally in the ratio \(m:n\). If the coordinates of the endpoints are A(x1, y1) and B(x2, y2), the coordinates x, y of the dividing point are given by:
x = frac{mx2 + nx1}{m + n}
y = frac{my2 + ny1}{m + n}
Section Formula for External Division
The section formula for external division finds the coordinates of a point dividing a line segment externally in the ratio \(m:n\). If the coordinates of the endpoints are A(x1, y1)\) and B(x2, y2), the coordinates x, y of the dividing point are given by:
x = frac{mx2 - nx1}{m - n}
y = frac{my2 - ny1}{m - n}
Application of Section Formula in Geometry
The section formula is crucial in geometry for solving problems related to dividing line segments, finding centroids, and working with coordinates. It provides a foundation for more advanced topics in coordinate geometry.
Importance of the Section Formula
The section formula is an essential tool in mathematics and real-life applications involving coordinate systems. Here are some key points about its importance:
It helps in dividing line segments in a specific ratio, aiding in construction and design tasks.
Understanding the section formula allows students to grasp advanced concepts in geometry and vector analysis.
It is used to find specific points such as centroids and in calculations involving distances and midpoints.
Tips and Tricks for Memorizing the Section Formula
Students often find the section formula tricky, but with some tips and tricks, it can be mastered.
Remember that the formula involves ratios, and the point divides the line segment either internally or externally.
Visual aids like diagrams can help in understanding and memorizing the formula.
Practice with real-life scenarios, such as dividing lengths in projects or maps, to reinforce understanding.
Common Mistakes and How to Avoid Them While Using the Section Formula
Students often make errors when applying the section formula. Here are some common mistakes and how to avoid them.
Mistake 1
Confusing Internal and External Division
Students may confuse the formulas for internal and external division. To avoid this, remember that internal division uses addition in the denominator, while external division uses subtraction.
Mistake 2
Incorrectly Applying Ratios
Errors can occur if the ratio is applied incorrectly. Always double-check which part of the line segment the ratio applies to and whether it is internal or external.
Mistake 3
Mixing Up Coordinates
When using the section formula, students might mix up the coordinates of the endpoints. To avoid this, label and organize the coordinates clearly before applying the formula.
Mistake 4
Forgetting to Simplify
Sometimes students forget to simplify the fractions in the formula. To avoid this, simplify the expressions in the formula after substituting the values.
Mistake 5
Ignoring Negative Signs
Students may ignore negative signs, especially in external division. Pay close attention to the signs when calculating the coordinates to ensure accuracy.
Examples of Problems Using the Section Formula
Problem 1
Find the point dividing the line segment joining (2, 3) and (10, 7) internally in the ratio 3:2.
The point is (6, 5)
Explanation
Using the section formula for internal division:
x = frac{3 × 10 + 2 × 2}{3 + 2} = frac{30 + 4}{5} = frac{34}{5} = 6.8
y = \frac{3 × 7 + 2 × 3}{3 + 2} = frac{21 + 6}{5} = frac{27}{5} = 5.4
Thus, rounding to the nearest integer, the point is (6, 5).
Problem 2
Find the coordinates of the point dividing the line segment joining (-1, 4) and (3, 8) externally in the ratio 1:3.
The point is (7, 10)
Explanation
Using the section formula for external division:
x = frac{1 × 3 - 3 × (-1)}{1 - 3} = frac{3 + 3}{-2} = -3
y = frac{1 × 8 - 3 × 4}{1 - 3} = frac{8 - 12}{-2} = 2
Thus, the coordinates of the point are (7, 10).
Problem 3
Determine the point dividing the line segment from (5, -2) to (15, 4) internally in the ratio 2:3.
The point is (11, 2)
Explanation
Using the section formula for internal division:
x = frac{2 × 15 + 3 × 5}{2 + 3} = frac{30 + 15}{5} = frac{45}{5} = 9
y = frac{2 × 4 + 3 × (-2)}{2 + 3} = frac{8 - 6}{5} = frac{2}{5} = 0.4
Thus, rounding to the nearest integer, the point is (11, 2).
Problem 4
Find the point dividing the segment joining (0, 0) and (6, 8) externally in the ratio 2:1.
The point is (12, 16)
Explanation
Using the section formula for external division:
x = frac{2 × 6 - 1 × 0}{2 - 1} = frac{12}{1} = 12
y = \frac{2 × 8 - 1 × 0}{2 - 1} = frac{16}{1} = 16
Thus, the coordinates of the point are (12, 16).
FAQs on the Section Formula
1.What is the section formula for internal division?
2.What is the section formula for external division?
3.How is the section formula used in geometry?
4.Can the section formula be used for three-dimensional coordinates?
5.What is a practical application of the section formula?
Glossary for Section Formula
- Section Formula: A mathematical formula used to determine the coordinates of a point dividing a line segment in a given ratio.
- Internal Division: Dividing a line segment between two points within the segment.
- External Division: Dividing a line segment beyond its endpoints.
- Ratio: A relationship between two numbers indicating how many times the first number contains the second.
- Coordinate Geometry: A branch of geometry where the position of points is defined using coordinates.
Jaskaran Singh Saluja
About the Author
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
Fun Fact
: He loves to play the quiz with kids through algebra to make kids love it.
