Grade 9
  1. Number Systems  1.1. Real Numbers  1.2. Rational Numbers  1.3. Irrational Numbers  1.4. Properties of Real Numbers  1.5. Laws of Exponents and Surds  1.6. Representation on the Number Line  1.7. Decimal Representation of Real Numbers
  2. Polynomials  2.1. Definition and Types of Polynomials  2.2. Degree of a Polynomial  2.3. Zeroes of a Polynomial  2.4. Remainder Theorem  2.5. Factorization of Polynomials  2.6. Algebraic Identities  2.7. Polynomial Equations and Roots
  3. Coordinate Geometry  3.1. Cartesian System  3.2. Plotting Points on the Plane  3.3. Distance Formula  3.4. Section Formula  3.5. Area of a Triangle  3.6. Coordinates of Midpoint and Centroid
  4. Linear Equations in Two Variables  4.1. Solutions of a Linear Equation  4.2. Graphical Method  4.3. Algebraic Method  4.4. Applications of Linear Equations  4.5. Linear Inequalities
  5. Introduction to Euclidean Geometry  5.1. Basic Definitions and Terms  5.2. Axioms and Postulates  5.3. Theorems on Angles and Lines  5.4. Properties of Parallel Lines  5.5. Construction of Triangles  5.6. Congruence Axioms
  6. Lines and Angles  6.1. Angle Pair Relationships  6.2. Parallel Lines and Transversal  6.3. Properties of Angles Formed by Parallel Lines  6.4. Angle Sum Property of a Triangle  6.5. Types of Angles
  7. Triangles  7.1. Congruence of Triangles  7.2. Criteria for Congruence  7.3. Properties of Triangles  7.4. Inequalities in a Triangle  7.5. Similarity of Triangles  7.6. Pythagorean Theorem
  8. Quadrilaterals  8.1. Properties of Quadrilaterals  8.2. Types of Quadrilaterals  8.3. Midpoint Theorem  8.4. Properties of Parallelograms  8.5. Trapezium and Kite Properties  8.6. Diagonals and their Properties
  9. Areas of Parallelograms and Triangles  9.1. Area of a Parallelogram  9.2. Area of a Triangle  9.3. Proofs of Area Theorems  9.4. Applications of Area Theorems
  10. Circles  10.1. Definitions and Terms  10.2. Properties of a Circle  10.3. Chord Properties  10.4. Tangents to a Circle  10.5. Cyclic Quadrilaterals  10.6. Arcs and Angles Subtended
  11. Constructions  11.1. Basic Constructions  11.2. Constructing Bisectors  11.3. Constructing Triangles  11.4. Constructing Tangents to a Circle  11.5. Constructing Angles of Given Measure
  12. Heron's Formula  12.1. Area of a Triangle Using Heron’s Formula  12.2. Application to Different Triangles and Quadrilaterals  12.3. Proof of Heron’s Formula
  13. Surface Areas and Volumes  13.1. Surface Area of a Cube Cuboid and Cylinder  13.2. Volume of a Cube Cuboid and Cylinder  13.3. Surface Area and Volume of a Cone  13.4. Surface Area and Volume of a Sphere and Hemisphere  13.5. Conversion of Units  13.6. Volume of a Prism
  14. Statistics  14.1. Collection of Data  14.2. Presentation of Data  14.3. Measures of Central Tendency  14.4. Mean Median and Mode  14.5. Graphical Representation of Data  14.6. Range and Measures of Dispersion
  15. Probability  15.1. Introduction to Probability  15.2. Experimental Probability  15.3. Theoretical Probability  15.4. Simple Problems on Probability

Grade 9 → Coordinate Geometry ↓


Section Formula


The section formula in coordinate geometry is a method used to find the coordinates of a point that divides a line segment into two parts in a given ratio. It is a useful tool in mathematics for solving problems related to dividing a line segment in a specific way. Let us look at this concept in detail with detailed examples and mathematical derivations.

Understanding line segments

A line segment is a part of a line that has two end points. Imagine a straight line drawn between these end points. If you want to find a specific point along this line that divides it in a ratio of 2:3, how would you do that? This is where the section formula comes into play.

Types of partitions

The segment formula can be used for two types of division:

  • Internal divisions
  • External divisions

Internal divisions

Internal division means that the point dividing the line segment lies between the two endpoints.

Finding the split point internally

Suppose you have a line segment defined by two points, point A(x 1, y 1) and point B(x 2, y 2). If point P divides line segment AB internally in the ratio m:n, then the coordinates of point P can be calculated using the following section formula:

    P(x, y) = ( (mx2 + nx1) / (m + n), (my2 + ny1) / (m + n) )

In the above formula:

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

Example of interior partition

Let us take an example to understand this concept better.

Example 1:

Suppose you have a line segment with end points A(2, 3) and B(8, 5). Find the point that divides the line segment internally in the ratio of 3:2.

By using the section formula:

    P(x, y) = ( (3 * 8 + 2 * 2) / (3 + 2), (3 * 5 + 2 * 3) / (3 + 2) ) = ( (24 + 4) / 5, (15 + 6) / 5 ) = (28/5, 21/5)

Thus, the coordinates of the point dividing the line segment in the ratio of 3:2 are (28/5, 21/5).

Visual example

A(2, 3) B(8, 5) P(28/5, 21/5)

External divisions

External division means that the point dividing the line segment does not lie between the two endpoints, but extends outside the line segment.

Detecting split points externally

When point P divides line segment AB externally in the ratio m:n, you use the segment formula a little differently:

    P(x, y) = ( (mx2 - nx1) / (m - n), (my2 - ny1) / (m - n) )

In the above formula:

  • x = ( mx2 - nx1 ) / (m - n)
  • y = (m y 2 - ny 1) / (m - n)

Example of external division

Let's look at an example to see how this works.

Example 2:

Consider a line segment with endpoints A(1, 2) and B(5, 6). Find the point which divides the line segment externally in the ratio 1:2.

Applying the section formula to the external division:

    P(x, y) = ( (1 * 5 - 2 * 1) / (1 - 2), (1 * 6 - 2 * 2) / (1 - 2) ) = ( (5 - 2) / -1, (6 - 4) / -1 ) = (3/-1, 2/-1) = (-3, -2)

Thus, the coordinates of the point dividing the line segment externally in the ratio 1:2 are (-3, -2).

Visual example

A(1, 2) B(5, 6) P(-3, -2)

Special cases

The section formula is not limited to just finding some arbitrary points that divide a line segment. There are special cases where it can be applied to specific geometric relationships, such as finding midpoints or centroids.

Middle

The midpoint divides a line segment into two equal parts. The midpoint can be found by setting m:n to a ratio of 1:1. This simplifies the formula to a very simple form:

    Midpoint M(x, y) = ( (x1 + x2) / 2, (y1 + y2) / 2 )

Example:

To find the midpoint of a line segment with end points A(4, 8) and B(10, 6).

    Midpoint M(x, y) = ( (4 + 10) / 2, (8 + 6) / 2 ) = (14/2, 14/2) = (7, 7)

The midpoint of the line segment is (7, 7).

A(4, 8) B(10, 6) M(7, 7)

Practice problems

Now, solve these practice problems to solidify your understanding of the section formula:

  1. Find the point which divides internally the line segment joining the points A(-3, 4) and B(5, -2) in the ratio 2:3.
  2. Determine the point that bisects the line segment between coordinates A(0, 2) and B(10, -4).
  3. Find the external division point for a line segment whose end points A(2, 3) and B(6, 7) are in the ratio 5:3.
  4. Calculate the point which divides the line segment joining A(-7, -4) and B(3, 9) internally in the ratio 4 : 7 and externally in the ratio 3 : 2.

Conclusion

The section formula is a fundamental tool in coordinate geometry that helps find the exact point along a line that divides it in specified ratios. Whether internally or externally, the formula provides a structured approach to solving geometry problems involving line segments. By practicing and applying it in different scenarios, you can enhance your understanding and problem-solving skills in geometry.


Grade 9 → 3.4


U
username
0%
completed in Grade 9

← Prev (3.3)
Distance Formula
Next (3.5) →
Area of a Triangle

Comments 



Section Formula

https://www.buddymath.com/g9/3.4?bmal=en