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 ↓


Distance Formula


The distance formula in coordinate geometry helps to calculate the distance between two points in the plane. Originating from the Pythagorean theorem, it serves as a fundamental tool in mathematics, especially in algebra and geometry.

Understanding the basics

First, consider two points on the Cartesian plane: point A ((x1, y1)) and point B ((x2, y2)). The distance formula enables us to calculate the length of the line segment connecting these two points.

Formula

The mathematical expression for the distance d between points A and B is given as:

D = √((x2 - x1)² + (y2 - y1)²)

This formula is derived from the Pythagorean Theorem, which states that 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.

Visual representation

Imagine two points A and B on a 2D plane. Create a right-angled triangle by drawing horizontal and vertical lines through these points.

A (x1, y1) B (x2, y2)

Step-by-step calculation

Let us learn how to use the formula to find the distance between two points with coordinates.

Example 1

Find the distance between point A ((3, 4)) and point B ((7, 1)).

  1. Calculate the difference in the x-coordinates: x2 - x1 = 7 - 3 = 4.
  2. Calculate the difference in the y-coordinates: y2 - y1 = 1 - 4 = -3.
  3. Apply the distance formula: 
    D = √((4)² + (-3)²)
d = √(16 + 9)
d = √25
d = 5

The distance between two points is 5 units.

Example 2

Let's try another example with points C ((-2, 8)) and D ((4, -3)).

  1. Difference in x-coordinates: 4 - (-2) = 4 + 2 = 6.
  2. Difference in y-coordinate: -3 - 8 = -11.
  3. Distance calculation: 
    D = √((6)² + (-11)²)
D = √(36 + 121)
d = √157
d ≈ 12.53

The approximate distance between these points is 12.53 units.

Applying usage to the real world

The distance formula is not limited to pure mathematics. It is widely used in physics, navigation and other fields.

Physics

In physics, this formula is essential for dealing with vectors, motion problems, and spatial calculations. For example, determining the shortest path between two positions.

Navigation

In navigation and GPS technology, the distance formula helps calculate the straight-line distance ("as-is") between two geographic locations.

Practice problems

Now it's your turn. Try applying the distance formula to solve the following problems:

Problem 1

Calculate the distance between the points E ((5, 9)) and F ((-3, 6)).

Problem 2

Find the distance between point G ((0, 0)) and point H ((-6, 8)).

Solution

Solve these questions and check your answers:

Solution 1

For problem 1:

D = √((-3 - 5)² + (6 - 9)²)
D = √((-8)² + (-3)²)
D = √(64 + 9)
d = √73
d ≈ 8.54

The distance is approximately 8.54 units.

Solution 2

For problem 2:

D = √((-6 - 0)² + (8 - 0)²)
D = √((6)² + (8)²)
D = √(36 + 64)
d = √100
d = 10

The distance is exactly 10 units.

Conclusion

The distance formula is an important mathematical tool that simplifies calculating the distance between two points in a 2D plane. Its applications extend beyond mathematics into a variety of scientific and real-world contexts, underpinning navigation systems, physics calculations, and more. Mastering this formula not only enriches one's mathematical toolkit but also enhances the understanding of spatial relationships in everyday life.


Grade 10 → 3.2


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