Class 9 Mathematic Coordinate Geometry All Formula & Concept Notes | Formula Sheet

Formulas and concepts in Coordinate Geometry for Class 9 Mathematics:

1. Basics of Coordinate Geometry

• Coordinate Plane: The coordinate plane consists of two perpendicular lines:
  • X-axis: The horizontal axis.
  • Y-axis: The vertical axis.
• Origin: The point where the X-axis and Y-axis intersect, denoted as $(0, 0)$.
• Coordinates: Any point on the plane is represented by an ordered pair $(x, y)$, where:
  • $x$ is the abscissa (horizontal distance from the Y-axis).
  • $y$ is the ordinate (vertical distance from the X-axis).

2. Distance Formula

• Purpose: To find the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$.
• Formula: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$
  

3. Section Formula

• Purpose: To find the coordinates of a point $P(x, y)$ dividing a line segment joining points $A(x_1, y_1)$ and $B(x_2, y_2)$ in a given ratio $m:n$.
• Formula: $$P(x, y) = \left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right)$$
• Example: Find the point dividing the line segment joining $A(1, 2)$ and $B(4, 6)$ in the ratio 2:3. $$P(x, y) = \left(\frac{2 \times 4 + 3 \times 1}{2+3}, \frac{2 \times 6 + 3 \times 2}{2+3}\right) = \left(\frac{8 + 3}{5}, \frac{12 + 6}{5}\right) = \left(\frac{11}{5}, \frac{18}{5}\right)$$

4. Midpoint Formula

• Purpose: To find the midpoint of the line segment joining two points $A(x_1, y_1)$ and $B(x_2, y_2)$.
• Formula: $$M\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$
• Example: Find the midpoint of the segment joining $A(2, 3)$ and $B(4, 5)$. $$M\left(\frac{2 + 4}{2}, \frac{3 + 5}{2}\right) = \left(\frac{6}{2}, \frac{8}{2}\right) = (3, 4)$$

5. Area of a Triangle

• Purpose: To find the area of a triangle formed by three points $A(x_1, y_1)$, $B(x_2, y_2)$, and $C(x_3, y_3)$.
• Formula: $$\text{Area} = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right|$$
• Example: Find the area of a triangle with vertices $A(1, 2)$, $B(4, 5)$, and $C(6, 7)$. $$\text{Area} = \frac{1}{2} \left| 1(5-7) + 4(7-2) + 6(2-5) \right| = \frac{1}{2} \left| 1(-2) + 4(5) + 6(-3) \right| = \frac{1}{2} \left| -2 + 20 - 18 \right| = \frac{1}{2} \times 0 = 0 \text{ square units}$$ (The area is 0, meaning the points are collinear.)

6. Concept of Collinearity

• Collinear Points: Three or more points are collinear if they lie on the same straight line.
• Condition: If the area of the triangle formed by three points is 0, the points are collinear.

7. Slope of a Line (Not in Class 9, but Foundational for Higher Studies)

• Slope: The slope of a line through two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by: $$m = \frac{y_2 - y_1}{x_2 - x_1}$$
• Example: The slope of the line passing through $A(2, 3)$ and $B(4, 5)$ is: $$m = \frac{5 - 3}{4 - 2} = \frac{2}{2} = 1$$

8. Concept of Quadrants

• Quadrants: The coordinate plane is divided into four quadrants:
  • Quadrant I: $x > 0$ and $y > 0$
  • Quadrant II: $x < 0$ and $y > 0$
  • Quadrant III: $x < 0$ and $y < 0$
  • Quadrant IV: $x > 0$ and $y < 0$

9. Applications of Coordinate Geometry

• Graphing Equations: Plotting points and graphing linear equations.
• Finding Distances: Measuring distances between points.
• Finding Midpoints: Identifying the middle point between two locations.
• Dividing Line Segments: Finding the precise point of division of line segments.

These are the essential formulas and concepts in Coordinate Geometry for Class 9. Mastering these will help you in solving problems and understanding more complex topics in higher classes.