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 ↓


Cartesian System


The Cartesian system is a fundamental concept in coordinate geometry, the part of mathematics that deals with defining and representing geometric shapes in a numerical format. Named after the French mathematician René Descartes, the Cartesian system is a coordinate system that establishes a set of rules for indicating the exact location of points on a plane using an ordered pair of numbers. It forms the basis of analytical geometry and is important in connecting algebra and geometry.

Basics of the Cartesian system

The Cartesian coordinate system is created using two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). These axes intersect at a point called the origin, represented as (0, 0).

X Y (0,0)

Understanding coordinates

In the Cartesian coordinate system, each point on the plane is represented by an ordered pair (x, y), where:

  • x is the horizontal distance along the x-axis from the origin.
  • y is the vertical distance along the y-axis from the origin.

These coordinates can be positive or negative depending on the quadrant in which the point is located.

Quadrants of the Cartesian plane

The Cartesian plane is divided into four regions, called quadrants, and numbered in counterclockwise order starting on the positive side of the x-axis:

  • Quadrant I: Both x and y are positive.
  • Quadrant II: x is negative, y is positive.
  • Quadrant III: Both x and y are negative.
  • Fourth quadrant: x is positive, y is negative.
Quadrant I Quadrant II Quadrant III fourth quadrant (3, 2) (-3, 2) (-3, -2) (3, -2)

Note how the sign of x and y changes in each quadrant. This information is important for interpreting the location of points on the Cartesian plane.

Drawing points on the Cartesian plane

Plotting a point on the Cartesian plane means drawing it at a location specified by its coordinates. This involves using x and y values to determine the exact position of the point.

Example: Plotting points

Example: Plot the following points on the Cartesian plane: (4, 3), (-2, 5), (-3, -4) and (5, -6).

(4, 3) (-2, 5) (-3, -4) (5, -6)

The coordinates determine the position of each point as follows:

  1. (4, 3) is 4 units to the right of the origin on the x-axis and 3 units up on the y-axis, which places it in quadrant I.
  2. (-2, 5) is 2 units to the left and 5 units up from the origin, which places it in quadrant II.
  3. (-3, -4) is 3 units up and 4 units down, which places it in quadrant III.
  4. (5, -6) is 5 units to the right and 6 units down, which places it in the fourth quadrant.

Equations in the Cartesian system

Equations can represent geometric shapes on the Cartesian plane. The simplest form is the equation of a line, usually given in slope-intercept form y = mx + c, where m is the slope and c is the y-intercept.

Example: Equation of a line

Example: Consider the line equation y = 2x + 1.

To graph this line, choose some values for x and calculate the corresponding y value:

x | y ------ 0 | 1 (y = 2*0 + 1) 1 | 3 (y = 2*1 + 1) 2 | 5 (y = 2*2 + 1)
(0, 1) (1, 3) (2, 5)

Notice how the line passes through the points (0, 1), (1, 3) and (2, 5), confirming its equation y = 2x + 1.

Distance formula

The distance formula is used to calculate the distance between two points on the Cartesian plane. Given two points (x1, y1) and (x2, y2), the distance d between them is calculated as:

d = √((x2 - x1) 2 + (y2 - y1) 2)

Example: Calculating distance

Example: Calculate the distance between (2, 3) and (5, 7).

Putting the values in the distance formula:

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

Therefore, the distance between (2, 3) and (5, 7) is 5 units.

Midpoint formula

The midpoint formula is used to find the midpoint of a line segment on the Cartesian plane. The midpoint M of the line segment joining two points (x1, y1) and (x2, y2) is given by:

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

Example: Finding the midpoint

Example: Find the midpoint of the line segment joining (3, 4) and (7, 8).

Use of midpoint formula:

M = ((3 + 7)/2, (4 + 8)/2) M = (10/2, 12/2) M = (5, 6)

Thus, the midpoint of the line segment is (5, 6).

Slope of the line

The slope of a line measures its steepness and direction. It is calculated as the ratio of the change in y to the change in x between two points on the line. Given the points (x1, y1) and (x2, y2), the slope m is:

m = (y2 - y1) / (x2 - x1)

Example: Calculating slope

Example: Determine the slope of the line passing through (1, 2) and (4, 6).

Applying the slope formula:

m = (6 - 2) / (4 - 1) m = 4 / 3

The slope of the line is 4/3.

Conclusion

The Cartesian system is an essential framework in coordinate geometry that enables us to locate points, construct shapes, and understand geometric relationships algebraically. By investigating coordinates, calculating distances, finding midpoints, and determining slopes, the Cartesian system forms the basis for analytical geometry and a variety of applications in science and engineering. Understanding this system helps solve complex problems and is fundamental in moving toward more advanced mathematical concepts.


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