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Mathematics Class 10 - Coordinate Geometry Notes

Comprehensive study notes for Class 10 - Coordinate Geometry olympiad preparation

Coordinate Geometry

Welcome to the chapter on Coordinate Geometry for Class 10. In this chapter, you will learn about the Cartesian plane, plotting points, distance formula, section formula, and their applications. By the end of this chapter, you will be able to solve problems involving points, lines, and distances on the coordinate plane.

Key Concepts

  • Coordinate Geometry: The study of geometry using the coordinate plane.
  • Cartesian Plane: A plane defined by two perpendicular axes: the x-axis (horizontal) and y-axis (vertical).
  • Origin: The point (0, 0) where the x-axis and y-axis intersect.
  • Coordinates: An ordered pair (x, y) that shows the position of a point on the plane.

Plotting Points

To plot a point, start at the origin. Move x units along the x-axis and y units along the y-axis.

  • Example: (3, 2) means 3 units right and 2 units up from the origin.
  • (-4, -1) means 4 units left and 1 unit down from the origin.

Quadrants

  • The coordinate plane is divided into four quadrants:
  • Quadrant I: (+, +)
  • Quadrant II: (-, +)
  • Quadrant III: (-, -)
  • Quadrant IV: (+, -)

Distance Formula

The distance between two points (x1, y1) and (x2, y2) is:

Distance = √[(x2 - x1)² + (y2 - y1)²]

  • Example: Find the distance between (1, 2) and (4, 6):
    Distance = √[(4-1)² + (6-2)²] = √[9 + 16] = √25 = 5 units

Section Formula

The section formula gives the coordinates of a point dividing the line segment joining (x1, y1) and (x2, y2) in the ratio m:n:

(x, y) = ( (mx2 + nx1)/(m+n), (my2 + ny1)/(m+n) )

  • If the point divides the segment internally, use the formula as above.
  • If externally, use (mx2 - nx1)/(m-n), (my2 - ny1)/(m-n).

Area of a Triangle (Using Coordinates)

The area of a triangle with vertices at (x1, y1), (x2, y2), and (x3, y3) is:

Area = ½ | x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2) |

Practice Questions

  1. Plot the points (2, 3), (-1, 4), and (0, -2) on a Cartesian plane.
  2. Find the distance between the points (5, 1) and (1, 4).
  3. Find the coordinates of the point dividing the line joining (2, 3) and (8, 7) in the ratio 2:1.
  4. Calculate the area of the triangle with vertices (1, 1), (4, 1), and (4, 5).
  5. In which quadrant does the point (-3, -5) lie?

Challenge Yourself

  • Show that the points (1, 2), (4, 6), and (7, 10) are collinear.
  • Find the area of a triangle with vertices (0, 0), (a, 0), and (0, b).

Did You Know?

  • Coordinate geometry was developed by René Descartes, a French mathematician.
  • It is used in computer graphics, navigation, and engineering!

Glossary

  • Coordinate: A pair of numbers showing the position of a point.
  • Quadrant: One of the four regions of the Cartesian plane.
  • Origin: The point (0, 0) on the coordinate plane.
  • Collinear: Points lying on the same straight line.

Answers to Practice Questions

  1. Points plotted as per their coordinates.
  2. Distance = √[(5-1)² + (1-4)²] = √[16 + 9] = √25 = 5 units
  3. Coordinates = (2×8 + 1×2)/(2+1), (2×7 + 1×3)/(2+1) = (16+2)/3, (14+3)/3 = (18/3, 17/3) = (6, 5.67)
  4. Area = ½ |1(1-5) + 4(5-1) + 4(1-1)| = ½ |-4 + 16 + 0| = ½ × 12 = 6 units²
  5. Quadrant III

Practice plotting points and using formulas to master coordinate geometry!

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