Grade 11
  1. Algebra  1.1. Sequences and Series  1.1.1. Arithmetic Sequence  1.1.2. Arithmetic Series  1.1.3. Geometric Sequence  1.1.4. Geometric Series  1.1.5. Convergence and Divergence  1.1.6. Sum to Infinity  1.1.7. Sigma Notation  1.2. Complex Numbers  1.2.1. Imaginary and Complex Numbers  1.2.2. Operations with Complex Numbers  1.2.3. Polar and Cartesian Forms  1.2.4. Complex Conjugates  1.2.5. Modulus and Argument  1.2.6. De Moivre's Theorem  1.2.7. Powers and Roots of Complex Numbers  1.3. Binomial Theorem  1.3.1. Expansion of Binomials  1.3.2. General Term  1.3.3. Binomial Coefficients  1.3.4. Applications of Binomial Theorem  1.3.5. Pascal's Triangle
  2. Functions and Graphs  2.1. Types of Functions  2.1.1. Polynomial Functions  2.1.2. Rational Functions  2.1.3. Exponential Functions  2.1.4. Logarithmic Functions  2.1.5. Trigonometric Functions  2.1.6. Piecewise Functions  2.2. Transformations of Functions  2.2.1. Translations  2.2.2. Reflections  2.2.3. Dilations and Stretches  2.2.4. Rotations  2.3. Inverse Functions  2.3.1. Finding Inverses  2.3.2. Graphs of Inverse Functions  2.3.3. Properties of Inverse Functions  2.3.4. Restrictions for Inverses
  3. Trigonometry  3.1. Trigonometric Ratios and Identities  3.1.1. Basic Trigonometric Ratios  3.1.2. Pythagorean Identities  3.1.3. Sum and Difference Formulas  3.1.4. Double Angle Formulas  3.1.5. Half Angle Formulas  3.1.6. Product-to-Sum and Sum-to-Product Formulas  3.2. Trigonometric Equations  3.2.1. Solving Trigonometric Equations  3.2.2. General Solutions in Trigonometric Equations  3.3. Graphs of Trigonometric Functions  3.3.1. Sine Graph  3.3.2. Cosine Graph  3.3.3. Tangent Graph  3.3.4. Amplitude and Period Adjustments  3.3.5. Phase Shifts in Graphs of Trigonometric Functions  3.4. Applications of Trigonometry  3.4.1. Laws of Sines and Cosines  3.4.2. Area of Triangles  3.4.3. Solving Triangles  3.4.4. Bearings and Navigation
  4. Calculus  4.1. Limits and Continuity  4.1.1. Concept of a Limit  4.1.2. One-Sided Limits  4.1.3. Limits at Infinity  4.1.4. Continuity of a Function  4.1.5. Types of Discontinuities  4.2. Differentiation  4.2.1. Definition of Derivative  4.2.2. Rules of Differentiation  4.2.3. Higher Order Derivatives  4.2.4. Chain Rule  4.2.5. Product Rule  4.2.6. Quotient Rule  4.3. Applications of Differentiation  4.3.1. Rate of Change  4.3.2. Tangents and Normals  4.3.3. Maxima and Minima  4.3.4. Increasing and Decreasing Functions  4.3.5. Optimization Problems  4.3.6. Curve Sketching  4.3.7. Related Rates  4.4. Integration  4.4.1. Antiderivatives  4.4.2. Indefinite Integrals  4.4.3. Definite Integrals  4.4.4. Fundamental Theorem of Calculus  4.4.5. Techniques of Integration  4.4.6. Area Under a Curve  4.5. Applications of Integration  4.5.1. Area Between Curves  4.5.2. Volumes of Solids of Revolution  4.5.3. Average Value of a Function
  5. Vectors and Matrices  5.1. Vectors  5.1.1. Representation of Vectors  5.1.2. Vector Operations  5.1.3. Dot Product  5.1.4. Cross Product  5.1.5. Applications in Geometry  5.1.6. Projection of Vectors  5.2. Matrices  5.2.1. Types of Matrices  5.2.2. Matrix Operations  5.2.3. Determinants  5.2.4. Inverse of a Matrix  5.2.5. Solving Systems of Linear Equations  5.2.6. Cramer's Rule
  6. Probability and Statistics  6.1. Probability  6.1.1. Basic Probability Concepts  6.1.2. Conditional Probability  6.1.3. Independent and Dependent Events  6.1.4. Bayes' Theorem  6.1.5. Combinatorial Probability  6.2. Random Variables  6.2.1. Discrete Random Variables  6.2.2. Continuous Random Variables  6.2.3. Probability Distributions  6.3. Probability Distributions  6.3.1. Binomial Distribution  6.3.2. Normal Distribution  6.3.3. Poisson Distribution  6.3.4. Uniform Distribution  6.4. Statistics  6.4.1. Measures of Central Tendency  6.4.2. Measures of Dispersion  6.4.3. Data Collection and Representation  6.4.4. Correlation and Regression  6.4.5. Sampling Techniques
  7. Coordinate Geometry  7.1. Straight Lines  7.1.1. Slope and Intercept Form  7.1.2. Distance Formula  7.1.3. Midpoint Formula  7.1.4. Angle Between Lines  7.1.5. Equation of Lines  7.2. Conic Sections  7.2.1. Circle  7.2.2. Parabola  7.2.3. Ellipse  7.2.4. Hyperbola  7.2.5. Properties of Conics  7.2.6. Parametric Equations of Conics  7.2.7. Polar Coordinates of Conics
  8. Mathematical Reasoning  8.1. Logic  8.1.1. Statements and Logical Connectives  8.1.2. Truth Tables  8.1.3. Logical Equivalence  8.1.4. Quantifiers  8.1.5. Proof Techniques  8.2. Proofs  8.2.1. Direct Proof  8.2.2. Indirect Proof  8.2.3. Proof by Contradiction  8.2.4. Proof by Mathematical Induction

Grade 11 → Coordinate Geometry → Straight Lines ↓


Distance Formula


Coordinate geometry is a branch of mathematics that helps us determine the position and size of geometric shapes using a coordinate system. One of the fundamental concepts in coordinate geometry is the distance formula. This formula allows us to find the distance between two points in a plane. Let's learn about the distance formula in detail and see how it can be applied to various problems.

Understanding the basics

Consider a plane with a coordinate system, commonly known as a Cartesian plane, where each point is given by a pair of coordinates (x, y). These coordinates indicate the position of the point along the X-axis and the Y-axis, respectively.

Distance between two points

If you have two points on this plane, say P1(x1, y1) and P2(x2, y2), then the distance between these two points can be calculated using the distance formula. The formula is derived from the Pythagorean theorem and is given as:

Distance = √((x2 - x1)² + (y2 - y1)²)

Here, √ represents the square root, (x2 - x1) is the difference in the X-coordinates, and (y2 - y1) is the difference in the Y-coordinates.

Derivation of the distance formula

Pythagorean theorem

The distance formula is essentially an application of the Pythagorean theorem. Consider a right triangle where the distance between two points is the hypotenuse. The other two sides are the difference in the X and Y coordinates.

p1(x1, y1) p2(x2, y2)

In the above triangle:

  • The horizontal side is |x2 - x1|.
  • The vertical side is |y2 - y1|.
  • The hypotenuse is the distance between points P1 and P2.

Therefore, by the Pythagorean theorem:

Hypotenuse² = (Base)² + (Perpendicular)² ⇒ (Distance)² = (x2 - x1)² + (y2 - y1)² ⇒ Distance = √((x2 - x1)² + (y2 - y1)²)

Visualizing with examples

Example 1

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

Using the distance formula:

Distance = √((7 - 3)² + (1 - 4)²) Distance = √((4)² + (-3)²) Distance = √(16 + 9) Distance = √25 Distance = 5

The distance between points A and B is 5 units.

A(3, 4) B(7, 1)

Example 2

Find the distance between the points C(0, 0) and D(8, 6).

Using the distance formula:

Distance = √((8 - 0)² + (6 - 0)²) Distance = √(8² + 6²) Distance = √(64 + 36) Distance = √100 Distance = 10

The distance between points C and D is 10 units.

C(0, 0) D(8, 6)

Practical applications of distance formula

The distance formula is not limited to theoretical problems only. It has many practical applications in real life scenarios:

1. Navigation and mapping

The distance formula is widely used in navigation systems to calculate the shortest path or direct distance between two locations determined by coordinates (latitude and longitude).

2. Computer graphics

In computer graphics, calculating the distance between points helps render 3D objects, determine object boundaries, and detect collisions in video games.

3. Robotics

Robots use distance formulas to navigate their surrounding environments, particularly in path-finding algorithms where they need to move efficiently from one point to another.

Increase understanding with practice problems

Now that we have learned the distance formula and its applications, practicing with additional problems will help solidify the understanding.

Problem 1

Find the distance between the points E(1, 2) and F(4, 6).

Problem 2

If G(-3, 5) and H(2, -1) are two points on a plane, calculate the distance between them.

Problem 3

Determine the distance from the point I(2, 3) to the origin.

Use the given questions to test the practical application of distance formula. Calculate each step by substituting the values and sharpen your skills with more practice.

Conclusion

The distance formula is an important component of coordinate geometry and has wide implications in various fields of science and technology. It serves as a basic building block for more complex geometric concepts and is helpful in problem-solving. By understanding and practicing the distance formula, one can enhance geometric reasoning and effectively apply these principles in real-world scenarios.


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