Class 10 Notes - Coordinate Geometry

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Coordinate Geometry

Coordinate Geometry, also known as Cartesian Geometry, is the study of geometry using a coordinate system. It allows us to represent geometric shapes algebraically and solve geometric problems using equations and coordinates.

1. Cartesian Plane

  • The Cartesian plane (or coordinate plane) is formed by two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical).
  • The point where they intersect is called the origin (O), with coordinates (0, 0).
  • Any point on the plane is represented as an ordered pair (x, y).
  • The plane is divided into four quadrants.

2. Plotting Points

  • To plot a point (x, y), move x units along the x-axis and y units along the y-axis.
  • Example: To plot (3, 2), move 3 units right and 2 units up from the origin.

3. Distance Formula

The distance between two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is given by:

\( \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)

4. Section Formula

If a point P divides the line segment joining \( A(x_1, y_1) \) and \( B(x_2, y_2) \) in the ratio m:n, then the coordinates of P are:

\( P = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \)

5. Area of a Triangle

The area of a triangle with vertices \( A(x_1, y_1) \), \( B(x_2, y_2) \), and \( C(x_3, y_3) \) is:

\( \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \)

6. Applications

  • Finding distance between two points.
  • Dividing a line segment in a given ratio.
  • Calculating area of triangles and quadrilaterals.
  • Locating points and shapes on a plane.

7. Example Problems

  1. Plot the points (2, 3), (-1, 4), and (0, -2) on the Cartesian plane.
  2. Find the distance between the points (1, 2) and (4, 6).
    Solution: \( \sqrt{(4-1)^2 + (6-2)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \)
  3. Find the coordinates of the point that divides the line joining (2, -3) and (8, 5) in the ratio 1:2.
    Solution: \( \left( \frac{1 \times 8 + 2 \times 2}{1+2}, \frac{1 \times 5 + 2 \times (-3)}{1+2} \right) = (4, -\frac{1}{3}) \)
  4. Find the area of the triangle with vertices (1, 1), (4, 5), and (7, 2).

8. Summary

  • Coordinate geometry connects algebra and geometry using coordinates.
  • It helps in solving geometric problems algebraically.
  • Key formulas: Distance, Section, and Area of triangle.