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


Coordinate geometry, also known as analytical geometry, is a fascinating and powerful tool that combines algebraic principles with geometric concepts. It allows us to study geometry using coordinate systems and algebraic operations to describe and analyze geometric properties. This approach not only makes visual concepts computable but also opens up a new world of problem-solving strategies.

Understanding the coordinate plane

The starting point for coordinate geometry is the coordinate plane. The coordinate plane is essentially a grid used to locate points based on their position. It is made up of two perpendicular number lines:

  • The horizontal number line is known as the x-axis.
  • The vertical number line is called the y-axis.

These axes intersect at a point called the origin, denoted by (0, 0). The plane is divided into four quadrants:

Quadrant I : +,+ (both coordinates are positive)
Quadrant II : -,+ (x is negative, y is positive)
Quadrant III: -,- (both coordinates are negative)
Quadrant IV : +,- (x is positive, y is negative)
X Y (0,0)

In this coordinate plane, any point can be described using the ordered pair (x, y).

Plotting points

To plot a point on the coordinate plane, you need to determine x (horizontal) and y (vertical) values and find their intersection. Let's plot a few points together:

Example point:

  • A(3, 4): Start at the origin (0,0). Move 3 units to the right and 4 units up.
  • B(-2, 3): Move 2 units left and 3 units up from the origin.
  • C(0, 5): From the origin, stay at x = 0 (don't move left or right), and go up 5 units.
A(3,4) B(-2,3) C(0,5)

Distance between two points

An important aspect of coordinate geometry is determining the distance between two points. It can be found using the distance formula derived from the Pythagorean Theorem. The distance d between two points (x1, y1) and (x2, y2) is given by:

d = √((x2 - x1)² + (y2 - y1)²)

Let's calculate the distance between points A(3, 4) and B(-2, 3):

x1 = 3, y1 = 4, x2 = -2, y2 = 3

D = √((-2 - 3)² + (3 - 4)²)
  = √((-5)² + (-1)²)
  = √(25 + 1)
  = √26

Thus, the distance between point A and point B is √26.

Midpoint of a line segment

Another fundamental concept is to find the midpoint of a line segment connecting two points. The midpoint M of the line segment connecting the points (x1, y1) and (x2, y2) is given by the formula:

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

Let us find the midpoint between A(3, 4) and B(-2, 3):

x1 = 3, y1 = 4, x2 = -2, y2 = 3

mx = (3 + (-2)) / 2 = 1 / 2
my = (4 + 3) / 2 = 7 / 2

M = (0.5, 3.5)

The midpoint is M (0.5, 3.5).

Slope of the line

Slope measures the steepness or inclination of the line. In simple terms, it tells us how much the line is inclined. The slope m of a line through two points (x1, y1) and (x2, y2) is calculated by dividing the change in the y-coordinate by the change in the x-coordinate:

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

Finding the slope between A(3, 4) and B(-2, 3):

x1 = 3, y1 = 4, x2 = -2, y2 = 3

m = (3 - 4) / (-2 - 3)
  = (-1) / (-5)
  = 1/5

So the slope of the line passing through points A and B is 1/5.

Equation of line

Once you have determined the slope of a line, the second important task is to find its equation. The most familiar form is the slope-intercept form:

y = mx + c

Where: m = slope of the line c = y-intercept (the value of y when x = 0)

To find the equation of the line that passes through the point A(3, 4) with a slope of 1/5, we substitute 1/5 for m and use the coordinates of (x, y) to solve for c:

y = (1/5)x + c
4 = (1/5)(3) + c

c = 4 – 0.6
c = 3.4

Equation of the line: y = (1/5)x + 3.4

Thus, the equation of the line is y = (1/5)x + 3.4.

Applications of coordinate geometry

Coordinate geometry is not only important in mathematics but also plays a vital role in various applied fields such as physics, engineering, computer graphics, and navigation. Let us explore some real-life applications:

  • In navigation: Coordinate geometry helps plot courses and determine distances. GPS technology relies heavily on these principles.
  • In physics: It is used to describe phenomena such as the trajectories of particles and calculations involving forces.
  • In architecture: Floor plans and elevations can be designed and analyzed using coordinate geometry.
  • In art and design: Figure designs and compositions make extensive use of the principles of coordinate geometry for symmetry and balance.

Understanding and mastering coordinate geometry paves the way for handling more complex multidimensional shapes and spaces in higher mathematics.

Conclusion

Coordinate geometry is a bridge between algebraic equations and geometric shapes. It gives us the power to translate geometric problems into algebraic equations that can be solved by manipulating them. By understanding fundamental concepts such as the coordinate plane, plotting points, calculating distances, finding midpoints, determining slopes, and writing the equation of a line, we unlock the ability to solve a wide range of geometric problems with confidence.

Embrace the logical beauty of coordinate geometry, and marvel at how it helps you understand both the natural and man-made worlds from a precise mathematical perspective.


Grade 9 → 3


U
username
0%
completed in Grade 9

← Prev (2.7)
Polynomial Equations and Roots
Next (3.1) →
Cartesian System

Comments 



Coordinate Geometry

https://www.buddymath.com/g9/3?bmal=en