Grade 10
  1. Number Systems  1.1. Real Numbers  1.1.1. Properties of Real Numbers  1.1.2. Rational and Irrational Numbers  1.1.3. Decimal Representation of Real Numbers  1.1.4. Operations on Real Numbers  1.1.5. Real Numbers on the Number Line  1.2. Euclid's Division Lemma  1.3. Prime Factorization  1.4. LCM and HCF  1.5. Exponents and Radicals  1.5.1. Laws of Exponents  1.5.2. Simplifying Radical Expressions  1.5.3. Scientific Notation
  2. Algebra  2.1. Polynomials  2.1.1. Definition and Types of Polynomials  2.1.2. Degree of a Polynomial  2.1.3. Zeros of a Polynomial  2.1.4. Factorization of Polynomials  2.1.5. Algebraic Identities  2.2. Linear Equations in Two Variables  2.2.1. Graph of Linear Equations  2.2.2. Solutions of Linear Equations  2.2.3. Application of Linear Equations in Word Problems  2.3. Quadratic Equations  2.3.1. Standard Form of a Quadratic Equation  2.3.2. Methods of Solving Quadratic Equations  2.3.2.1. Factorization Method  2.3.2.2. Completing the Square Method  2.3.2.3. Quadratic Formula  2.3.3. Nature of Roots  2.4. Arithmetic Progressions  2.4.1. Definition of Arithmetic Progression  2.4.2. General Term of an AP  2.4.3. Sum of First n Terms of an AP  2.5. Introduction to Functions  2.5.1. Definition and Types of Functions  2.5.2. Domain and Range  2.5.3. Composition of Functions  2.5.4. Inverse of a Function
  3. Coordinate Geometry  3.1. Cartesian System  3.2. Distance Formula  3.3. Section Formula  3.4. Area of a Triangle  3.5. Slope of a Line  3.6. Equation of a Line  3.6.1. Slope-Intercept Form  3.6.2. Point-Slope Form  3.6.3. Two-Point Form  3.6.4. Intercept Form  3.6.5. General Form of a Line
  4. Trigonometry  4.1. Trigonometric Ratios  4.1.1. Sine Cosine Tangent and their Reciprocals  4.1.2. Values of Trigonometric Ratios at Specific Angles  4.2. Trigonometric Identities  4.3. Trigonometric Ratios of Complementary Angles  4.4. Heights and Distances  4.4.1. Angle of Elevation and Depression  4.4.2. Applications in Real-Life Problems
  5. Geometry  5.1. Triangles  5.1.1. Congruence of Triangles  5.1.2. Criteria for Congruence  5.1.3. Properties of Triangles  5.2. Similarity  5.2.1. Similar Figures  5.2.2. Criteria for Similarity  5.2.3. Similarity of Triangles  5.2.4. Applications of Similarity in Real-Life  5.3. Circles  5.3.1. Properties of a Circle  5.3.2. Tangent to a Circle  5.3.3. Number of Tangents from a Point  5.3.4. Angle Subtended by an Arc  5.4. Constructions  5.4.1. Construction of Similar Triangles  5.4.2. Tangent to a Circle  5.4.3. Construction of Angle Bisectors
  6. Mensuration  6.1. Areas of Plane Figures  6.1.1. Area of a Triangle  6.1.2. Area of a Parallelogram  6.1.3. Area of a Trapezium  6.1.4. Area of a Rhombus  6.2. Surface Areas and Volumes  6.2.1. Surface Area of a Cube Cuboid and Cylinder  6.2.2. Surface Area of a Cone and Sphere  6.2.3. Volume of a Cube Cuboid and Cylinder  6.2.4. Volume of a Cone and Sphere  6.3. Conversion of Units  6.4. Frustum of a Cone
  7. Statistics  7.1. Introduction to Statistics  7.2. Collection of Data  7.3. Presentation of Data  7.3.1. Tabular Form  7.3.2. Graphical Form  7.3.2.1. Bar Graph  7.3.2.2. Histogram  7.3.2.3. Frequency Polygon  7.3.2.4. Ogive  7.4. Measures of Central Tendency  7.4.1. Mean Median and Mode  7.4.2. Quartiles and Percentiles  7.5. Measures of Dispersion  7.5.1. Range and Interquartile Range  7.5.2. Standard Deviation and Variance
  8. Probability  8.1. Introduction to Probability  8.2. Experimental Probability  8.3. Theoretical Probability  8.4. Probability of Simple Events  8.5. Complementary Events  8.6. Probability of Compound Events

Grade 10 → Coordinate Geometry ↓


Section Formula


In coordinate geometry, the section formula is a very important concept that provides us with a way to find a particular point that divides a line segment in a specific ratio. This concept is commonly used in problems that involve dividing line segments in a specific way. Understanding the section formula is important in solving geometry problems related to dividing line segments. Let us now dive deeper into the concept of section formula.

Understanding line segments

Before we jump into the section formula, let's understand what a line segment is. A line segment is defined by two end points. For example, on a plane, consider a line segment formed by the points A(x 1, y 1) and B(x 2, y 2). This line segment is stable because it has an initial point A and an end point B.

What is section formula?

The section formula helps to find the coordinates of a point P(x, y) that divides a line segment between two given points A(x 1, y 1) and B(x 2, y 2) in a particular ratio m:n.

Section formula - Internal partition

When a point divides a line segment internally, the coordinates of the point P(x, y) are calculated using the internal division formula:

x = (mx 2 + nx 1)/(m + n) y = (my 2 + ny 1)/(m + n)

Here, the point P(x, y) divides the line segment AB in the ratio m:n.

Section formula - External division

Conversely, if the point divides the line segment externally, the formula changes slightly. The formula for the coordinates of the division point P(x, y) becomes:

x = (mx 2 - nx 1)/(m - n) y = (my 2 - ny 1)/(m - n)

Here, the point P(x, y) divides the line externally in the ratio m:n.

Viewing the section formula

Example: Internal division

Let us consider a line segment with end points A(2, 2) and B(8, 10). We have to find the point P which divides this line segment in the ratio 3:2.

Use of section formula for internal partitions:

x = (3*8 + 2*2)/(3 + 2) = (24 + 4)/5 = 28/5 = 5.6 y = (3*10 + 2*2)/(3 + 2) = (30 + 4)/5 = 34/5 = 6.8

Thus, the coordinates of point P are (5.6, 6.8).

(2, 2)(8, 10)(5.6, 6.8)

Example: External division

Consider the same points A(2, 2) and B(8, 10), but this time, let us find the point P which divides AB externally in the ratio 3:2.

Use of section formula for external division:

x = (3*8 - 2*2)/(3 - 2) = (24 - 4)/1 = 20 y = (3*10 - 2*2)/(3 - 2) = (30 - 4)/1 = 26

The coordinates of point P are (20, 26).

(2, 2)(8, 10)(20, 26)

Special cases in the section formula

Midpoint of a line segment

A special case of the section formula is when the ratio is 1:1. Here, point P is the midpoint of the line segment AB.

x = (x 1 + x 2)/2 y = (y 1 + y 2)/2

Using the same points A(2, 2) and B(8, 10) the midpoint is calculated as follows:

x = (2 + 8)/2 = 10/2 = 5 y = (2 + 10)/2 = 12/2 = 6

Hence, the midpoint of AB is (5, 6).

(2, 2)(8, 10)(5, 6)

Applications of section formula

The section formula is widely used in various fields of study including geometry, engineering, art design, robotics, and computer graphics. It helps in determining specific and necessary divisions of line segments in various applications.

Practical example

In civil engineering, builders often need to divide areas or line segments to make structures look correct and maintain proportions. For example, consider a scenario where an architect is designing a building. The building has corridors that must be divided into specific sections, with rooms proportionally spaced apart between the two end points. Using the section formula helps ensure that each section aligns precisely with the architectural plans.

Challenge yourself

Imagine a scenario where you need to plan a garden path with a diagonal of 70m in a rectangular park. Using internal partitions, the path needs to be divided equally into three parts by placing two benches at a distance based on the section formula. Using the end points of the diagonal, calculate where the benches should be placed.

The knowledge and understanding of section formulas provides a strong foundation to tackle similar complex calculation problems in other advanced areas of mathematics and geometry.


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Section Formula

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