Coordinate Geometry Class 10 ||Maths|| Chapter 7 NCERT Notes
Introduction to the Coordinate Plane:
The coordinate plane is a two-dimensional plane formed by the intersection of two perpendicular lines:
- The X-axis (horizontal line).
- The Y-axis (vertical line).
These axes divide the plane into four quadrants:
- Quadrant I: Both x and y are positive.
- Quadrant II: x is negative, y is positive.
- Quadrant III: Both x and y are negative.
- Quadrant IV: x is positive, y is negative.
Each point in this plane is defined by an ordered pair (x, y), called coordinates. The first number is the x-coordinate (or abscissa), and the second number is the y-coordinate (or ordinate).
Important Formulas in Coordinate Geometry:
1. Distance Formula:
The distance between two points and in a coordinate plane is given by the formula:
- This formula is derived from the Pythagoras Theorem in a right-angled triangle.
Application Example: Find the distance between the points and .
Using the formula:
2. Section Formula:
The section formula helps us find the coordinates of a point that divides a line segment joining two points in a given ratio.
- For internal division: If the point divides the line segment joining two points and in the ratio , then the coordinates of are:
- For external division: The formula remains the same, but the sign between and will change.
Application Example: Find the point that divides the line segment joining points and in the ratio 2:1 internally.
Using the section formula:
Thus, is the required point.
3. Midpoint Formula:
The midpoint of a line segment joining two points and can be found using the midpoint formula:
Application Example: Find the midpoint of the line segment joining and .
Using the formula:
Thus, the midpoint is .
4. Area of a Triangle:
The area of a triangle with vertices at points , , and is given by the formula:
- The absolute value ensures that the area is positive, regardless of the order of the points.
Application Example: Find the area of the triangle with vertices , , and .
Using the formula:
Thus, the area of the triangle is 8 square units.
Important Concepts to Remember:
Collinearity of Three Points:
If the area of a triangle formed by three points is zero, then the points are said to be collinear, i.e., they lie on the same straight line.Slope of a Line:
Though not a part of the main chapter in Class 10, it's helpful to know that the slope of a line joining two points and is:Quadrants Overview:
Each quadrant is determined by the sign of x and y coordinates:- Quadrant I: (+, +)
- Quadrant II: (-, +)
- Quadrant III: (-, -)
- Quadrant IV: (+, -)
Key Takeaways:
- Distance Formula helps to find the distance between two points in a plane.
- Section Formula and Midpoint Formula help in dividing a line segment in a given ratio or finding the midpoint of the segment.
- Area of a Triangle Formula is useful to determine the area when the vertices are known in coordinate form.
- Quick-thinking Tip: If you ever need to check the collinearity of points, calculate the area of the triangle. If it’s zero, they are collinear.