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 ↓


Slope of a Line


In coordinate geometry, the slope of a line is a number that measures the slope, direction, and inclination of the line. It is an important concept in algebra and geometry as it provides valuable information about the nature of lines on a plane. In this lesson, we will explore the concept of the slope of a line in depth and understand its importance in various contexts.

What is a slope?

The slope of a line is calculated by dividing the change in the y-coordinate by the change in the x-coordinate as you move along the line. Mathematically, the slope m of a line passing through two points (x₁, y₁) and (x₂, y₂) is given by the formula:

m = (y₂ - y₁) / (x₂ - x₁)

Here, y₂ - y₁ represents the change in the y-coordinate (also known as the rise), and x₂ - x₁ represents the change in the x-coordinate (also known as the run). The slope is essentially a ratio that indicates how much the y-coordinate changes for a one unit change in the x-coordinate.

Visual representation

Let's consider a visual example to understand the slope of a line more intuitively. Suppose you have a line that passes through the points (2, 3) and (5, 11). We can calculate the slope of this line using the formula discussed. First, let's plot the points in the coordinate system.

(2, 3) (5, 11)

In this diagram, the red points on the grid represent the coordinates (2, 3) and (5, 11). The blue line connecting them is the straight line whose slope we want to calculate. Using the slope formula:

m = (11 - 3) / (5 - 2) = 8 / 3

Thus, the slope of the line is 8/3, which means the line rises 8 units for every 3 units it moves horizontally.

Interpretation of slope

The slope of a line can describe several characteristics:

  • Positive slope: If the slope is positive, it means the line acts as an increasing function. As the x-coordinate increases, the y-coordinate also increases. The line slopes upward to the right.
  • Negative slope: If the slope is negative, it indicates a decreasing function. The y-coordinate decreases as the x-coordinate increases, and the line slopes downward to the right.
  • Zero slope: Zero slope implies that the line is horizontal, which indicates that there is no change in the y-coordinate when the x-coordinate changes.
  • Undefined slope: When a line is vertical, the change in the x-coordinate is zero, resulting in an undefined slope. Technically, division by zero is undefined, which is why vertical lines have an undefined slope.

Examples of different slopes

Positive slope

Example: Consider a line passing through the points (1, 2) and (3, 7).

m = (7 - 2) / (3 - 1) = 5 / 2
(1, 2) (3, 7)

A slope of 5/2 represents an upward inclination from left to right.

Negative slope

Example: Consider a line passing through the points (2, 6) and (4, 1).

m = (1 - 6) / (4 - 2) = -5 / 2
(2, 6) (4, 1)

A slope of -5/2 indicates a downward slope from left to right.

Zero slope

Example: The slope of the line passing through the points (1, 4) and (3, 4) is:

m = (4 - 4) / (3 - 1) = 0
(1, 4) (3, 4)

Zero slope indicates a flat horizontal line.

Undefined slope

Example: The slope of the line passing through the points (4, 2) and (4, -1) is undefined:

m = (-1 - 2) / (4 - 4) = undefined
(4, 2) (4, -1)

Undefined slope indicates a vertical line where the x-coordinates do not change.

Conclusion

Understanding the slope of a line is important in coordinate geometry because it tells us about the direction and slope of the line. By mastering the calculation and interpretation of slopes, you can understand more complex topics like linear equations, graphing lines, and real-world applications like rate of change in economics and physics.

Exploring slopes with more practice through visual examples and a variety of contexts will enrich your understanding and appreciation of this essential concept in mathematics.


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Slope of a Line

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