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 ↓


Distance Formula


In mathematics, coordinate geometry (also called analytic geometry) is the study of geometry using a coordinate system. The distance formula is an important aspect of this field. This formula calculates the distance between two points in a coordinate plane.

Introduction to the distance formula

The distance formula is derived from the Pythagorean theorem. In a two-dimensional plane, any two points can be represented as coordinates, say (x1, y1) and (x2, y2). The distance d between these points is given by:

d = √((x2 - x1)² + (y2 - y1)²)

Visual example

Let us illustrate this with a diagram. Suppose we have two points, A and B, with coordinates A(1, 2) and B(4, 6). We can see the distance between these points by plotting them on a grid.

A(1,2) b(4,6)

The distance d between A(1, 2) and B(4, 6) can be calculated using the distance formula:

d = √((4 - 1)² + (6 - 2)²) = √(3² + 4²) = √(9 + 16) = √25 = 5

Step-by-step explanation of the formula

Let's break down the distance formula to understand each part of it. The formula can be broken down into these parts:

  • (x2 - x1): This represents the difference in the x-coordinates of two points. It shows how far apart the points are horizontally.
  • (y2 - y1): This represents the difference in the y-coordinates, which gives the vertical separation of the points.
  • Squaring each of these differences ((x2 - x1)² and (y2 - y1)²) ensures that the differences become positive and, in essence, the separations become equal regardless of direction.
  • Adding the classes together allows these independently calculated distances to jointly contribute to the overall distance.
  • The square root satisfies the Pythagorean Theorem, and converts the squared input into a linear measure of distance.

Practical example

Example 1

Consider two points C(3, 3) and D(7, 8). We are given the task of finding the distance between these points.

d = √((7 - 3)² + (8 - 3)²) = √((4)² + (5)²) = √(16 + 25) = √41

Example 2

As another example, find the distance between the points E(-1, -1) and F(3, 3).

d = √((3 - (-1))² + (3 - (-1))²) = √((3 + 1)² + (3 + 1)²) = √(4² + 4²) = √(16 + 16) = √32 = 4√2

Explorable examples

Let's consider another situation. What if both points are on the x-axis or the y-axis?

Example 3

Evaluate the distance between the points G(0, 0) and H(0, 5).

d = √((0 - 0)² + (5 - 0)²) = √(0 + 25) = √25 = 5

Understanding through etymology

As explained earlier, the distance formula in coordinate geometry is conceptually connected to the Pythagorean Theorem: in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. Imagine a scenario in which the x-axis and y-axis form the legs of a right triangle. In this case, the horizontal segment associated with the difference in the x-coordinate and the vertical segment associated with the difference in the y-coordinate form the legs of the triangle. The line connecting our two points would then serve as the hypotenuse of the triangle.

Visual diagram

(x1, y1) (x2, y2)

Challenging yourself

Example 4

Determine the distance between the points I(6, 7) and J(-2, 3).

d = √((-2 - 6)² + (3 - 7)²) = √((-8)² + (-4)²) = √(64 + 16) = √80 = 4√5

Example 5

Now, what about the points K(-3, 5) and L(4, -1)?

d = √((4 - (-3))² + (-1 - 5)²) = √((4 + 3)² + (-6)²) = √(7² + 6²) = √(49 + 36) = √85

Applications in real life

The distance formula is not just an abstract mathematical equation, but it has practical applications in real life. For example, in navigation systems or mapping services, this formula often has to be used to find the shortest path from one place to another. Moreover, in fields such as computer graphics, physics simulations, and even analyzing data trends, understanding the distance between points can provide valuable insights.

Conclusion

By now, you should have a full understanding of the distance formula and how it serves as a powerful tool in coordinate geometry. By deriving its meaning from the Pythagorean Theorem, it gives us a systematic way to calculate the separation between points on a plane. Whether used in mathematics or applied in real-world scenarios, mastering this formula opens the way for exploration into countless geometric calculations in both theory and practice.


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Distance Formula

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