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Mathematics Class 9 - Coordinate Geometry Notes

Comprehensive study notes for Class 9 - Coordinate Geometry olympiad preparation

Coordinate Geometry

Welcome to the chapter on Coordinate Geometry for Class 9. In this chapter, you will learn about the Cartesian plane, plotting points, and finding the distance between points. By the end of this chapter, you will be able to use coordinates to solve geometry problems and understand how graphs work.

Key Concepts

  • Coordinate Geometry: The study of geometry using numbers and algebra on a graph.
  • Cartesian Plane: A plane with two perpendicular axes: the x-axis (horizontal) and y-axis (vertical).
  • Origin: The point (0, 0) where the x-axis and y-axis meet.
  • Coordinates: A pair of numbers (x, y) that show the position of a point on the plane.

Plotting Points

To plot a point, use its coordinates (x, y). Move x units along the x-axis, then y units along the y-axis.

  • Example: To plot (3, 2), move 3 units right and 2 units up from the origin.
  • The x-coordinate tells you how far to move horizontally.
  • The y-coordinate tells you how far to move vertically.

Quadrants

The Cartesian plane is divided into four quadrants:

  • Quadrant I: (+, +)
  • Quadrant II: (−, +)
  • Quadrant III: (−, −)
  • Quadrant IV: (+, −)

Distance Between Two Points

The distance between points (x1, y1) and (x2, y2) is found using the formula:

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 helps find the coordinates of a point dividing a line segment in a given ratio.

  • If point P divides AB in the ratio m:n, then
    P = ((mx2 + nx1)/(m+n), (my2 + ny1)/(m+n))

Applications

  • Locating points on maps and graphs
  • Finding distances and midpoints
  • Solving geometry problems using algebra

Practice Questions

  1. Plot the points (2, 3), (−1, 4), (−2, −3), and (3, −2) on a Cartesian plane.
  2. In which quadrant does the point (−5, 6) lie?
  3. Find the distance between (0, 0) and (6, 8).
  4. Find the midpoint of the line joining (2, 4) and (6, 8).
  5. If a point divides the line joining (1, 2) and (5, 6) in the ratio 1:1, what are its coordinates?

Challenge Yourself

  • Find the coordinates of a point dividing the line joining (3, 7) and (9, 1) in the ratio 2:3.
  • Draw a Cartesian plane and mark all four quadrants with examples.

Did You Know?

  • Coordinate geometry was invented by René Descartes, a French mathematician.
  • It helps connect algebra and geometry using graphs.

Glossary

  • Coordinate: A pair of numbers showing a point’s position.
  • Origin: The point (0, 0) on the Cartesian plane.
  • Quadrant: One of the four sections of the Cartesian plane.
  • Midpoint: The point exactly halfway between two points.

Answers to Practice Questions

  1. (2, 3): Quadrant I; (−1, 4): Quadrant II; (−2, −3): Quadrant III; (3, −2): Quadrant IV
  2. Quadrant II
  3. Distance = √[(6−0)² + (8−0)²] = √[36+64] = √100 = 10 units
  4. Midpoint = ((2+6)/2, (4+8)/2) = (4, 6)
  5. Coordinates = ((1+5)/2, (2+6)/2) = (3, 4)

Practice plotting points and using formulas to master coordinate geometry!

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