Coordinate Geometry Class 9

Coordinate Geometry – Study Material

Theory

The Cartesian plane, also known as the coordinate plane or rectangular coordinate system, is a two-dimensional surface defined by two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). These axes intersect at the point called the origin, denoted as (0, 0). The plane is divided into four regions called quadrants. Any point on this plane is identified by a pair of numerical coordinates (x, y), where x represents the horizontal distance and y the vertical distance from the origin.

This system was invented by the French mathematician René Descartes and forms the foundation for graphing algebraic equations and performing geometric transformations.

Solved Examples

  1. Plot the point (3, 2) on the Cartesian plane.
  2. Plot the point (-4, 5).
  3. Identify the quadrant of the point (-3, -6).
  4. What are the coordinates of the origin?
  5. Plot the point (0, -7). Is it on the x-axis or y-axis?

Exercise

  1. (2, 5)
  2. (-3, -4)
  3. (-5, 3)
  4. (4, -6)
  5. (0, 4)
  6. (0, 0)
  7. (6, -2)
  8. (-7, 0)

Theory

Each point in the Cartesian plane is represented by an ordered pair of numbers, known as coordinates. The first number in the pair is the x-coordinate (abscissa) and the second is the y-coordinate (ordinate). These numbers tell how far the point is from the origin along the x-axis and y-axis, respectively. A positive x-coordinate means the point is to the right of the y-axis; a negative one means to the left. Similarly, a positive y-coordinate indicates above the x-axis; a negative one indicates below.

Solved Examples

  1. Find the coordinates of the point that lies 3 units right and 2 units up from the origin.
  2. A point is 4 units left and 5 units down from the origin. What are its coordinates?
  3. Find the x- and y-coordinates of the origin.
  4. A point lies 6 units left and lies on the x-axis. What are its coordinates?
  5. What point lies on the y-axis and 7 units above the origin?

Exercise

  1. A point is 4 units right and 3 units down.
  2. A point lies at the intersection of x-axis and y-axis.
  3. A point is 6 units above and lies on y-axis.
  4. A point is 5 units left and 1 unit up.
  5. A point lies on x-axis, 2 units to the right.
  6. A point lies on x-axis, 3 units left.
  7. A point lies on y-axis, 6 units below.
  8. A point lies 0 units away from origin.

Theory

The Cartesian plane is divided into four quadrants by the x- and y-axes:

Quadrant Sign of x-coordinate Sign of y-coordinate
I (First) + +
II (Second) – +
III (Third) – –
IV (Fourth) + –

The signs help us identify the quadrant of any point. For instance, a point (3, -5) lies in the fourth quadrant because x is positive and y is negative.

Solved Examples

  1. Determine the quadrant of (5, 6).
  2. In which quadrant does (-4, 7) lie?
  3. Find the quadrant of (-3, -8).
  4. Identify the quadrant for (9, -2).
  5. Point (0, 6) lies on which axis?

Exercise

  1. (7, -9)
  2. (-6, 3)
  3. (0, -5)
  4. (3, 3)
  5. (-3, -2)
  6. (0, 0)
  7. (-1, 0)
  8. (8, 0)

To plot a point (x, y) in the Cartesian plane:

  • Start from the origin (0, 0).
  • Move x units along the x-axis: right if x is positive, left if negative.
  • From that position, move y units parallel to the y-axis: up if y is positive, down if negative.

This graphical representation helps understand functions, algebraic relationships, and geometry.

  1. Plot (1, 2).
  2. Plot (-2, 3).
  3. Plot (-3, -4).
  4. Plot (4, -1).
  5. Plot (0, 5).
  1. (2, -1)
  2. (-1, 1)
  3. (3, 4)
  4. (-4, -3)
  5. (0, -6)
  6. (5, 0)
  7. (0, 0)
  8. (-2, -5)
  1. What is the name of the point (0, 0) in the Cartesian plane?
    A. Base    B. Intersection    C. Origin    D. Center
  2. Which quadrant does the point (-5, 3) lie in?
    A. First    B. Second    C. Third    D. Fourth
  3. What is the x-coordinate of the point (4, -7)?
    A. 4    B. -7    C. 0    D. -4
  4. Which of the following points lies on the y-axis?
    A. (0, 4)    B. (4, 0)    C. (3, 3)    D. (-3, -3)
  5. A point in the third quadrant has:
    A. x > 0, y > 0    B. x < 0, y > 0    C. x < 0, y < 0    D. x > 0, y < 0
  6. Which axis does the point (-4, 0) lie on?
    A. x-axis    B. y-axis    C. z-axis    D. None
  7. The coordinates of a point 5 units right and 6 units down are:
    A. (5, 6)    B. (-5, -6)    C. (5, -6)    D. (-5, 6)
  8. The point (-7, -8) lies in which quadrant?
    A. I    B. II    C. III    D. IV
  9. Which point lies in the first quadrant?
    A. (3, -3)    B. (-3, 3)    C. (3, 3)    D. (-3, -3)
  10. What are the coordinates of a point lying on the origin?
    A. (1, 1)    B. (0, 1)    C. (1, 0)    D. (0, 0)
  11. If a point lies on the x-axis, its y-coordinate is:
    A. 1    B. 0    C. -1    D. Cannot be determined
  12. Which point lies in the fourth quadrant?
    A. (3, 4)    B. (3, -4)    C. (-3, 4)    D. (-3, -4)
  13. The ordinate of the point (5, 7) is:
    A. 5    B. 7    C. 0    D. 12
  14. A point has coordinates (-2, 0). It lies:
    A. On y-axis    B. On x-axis    C. In first quadrant    D. In second quadrant
  15. In which quadrant does the point (6, -9) lie?
    A. I    B. II    C. III    D. IV
  16. Which coordinate is also called the abscissa?
    A. x-coordinate    B. y-coordinate    C. Origin    D. Distance
  17. Which point is on the positive y-axis?
    A. (0, -3)    B. (3, 0)    C. (0, 3)    D. (-3, 0)
  18. The point (-1, 1) lies in which quadrant?
    A. I    B. II    C. III    D. IV
  19. What is the y-coordinate of the point (7, -2)?
    A. 7    B. -2    C. 0    D. -7
  20. Which of the following is true for any point on the x-axis?
    A. y = 0    B. x = 0    C. x = y    D. x = y = 0
  21. The point (0, -6) lies:
    A. On x-axis    B. In third quadrant    C. On y-axis    D. In fourth quadrant
  22. What is the name of the vertical axis?
    A. x-axis    B. y-axis    C. z-axis    D. t-axis
  23. In which quadrant does (-2, -3) lie?
    A. I    B. II    C. III    D. IV
  24. If a point lies on y-axis, what is its x-coordinate?
    A. 1    B. -1    C. 0    D. Cannot be determined
  25. Which point lies on both x-axis and y-axis?
    A. (0, 0)    B. (1, 1)    C. (0, 1)    D. (-1, 0)

Answers:

1. C | 2. B | 3. A | 4. A | 5. C | 6. A | 7. C | 8. C | 9. C | 10. D

11. B | 12. B | 13. B | 14. B | 15. D | 16. A | 17. C | 18. B | 19. B | 20. A

21. C | 22. B | 23. B | 24. C | 25. A

Coordinate Geometry: Self Assessment Paper

  1. The point (0, -5) lies on the:
    • A. x-axis
    • B. y-axis
    • C. origin
    • D. first quadrant
  2. The abscissa of (3, -7) is:
    • A. 3
    • B. -7
    • C. 0
    • D. -3
  3. In which quadrant is the point (6, -4)?
    • A. I
    • B. II
    • C. III
    • D. IV
  4. A point has coordinates (-3, 0), it lies on:
    • A. x-axis
    • B. y-axis
    • C. origin
    • D. third quadrant
  5. A point with negative x and positive y lies in:
    • A. I
    • B. II
    • C. III
    • D. IV
  6. Coordinates of origin are:
    • A. (1, 1)
    • B. (0, 1)
    • C. (1, 0)
    • D. (0, 0)
  7. The ordinate of the point (5, -9) is:
    • A. 5
    • B. -9
    • C. 0
    • D. -5
  8. Which axis is horizontal?
    • A. y-axis
    • B. x-axis
    • C. z-axis
    • D. t-axis
  1. Define the Cartesian plane. Mention its components.
  2. Identify the quadrant of the point (-4, -6) and explain.
  3. Write any two points lying on the x-axis.
  4. Write coordinates of a point which lies in the second quadrant and has ordinate 5.
  5. What are the coordinates of a point that is 6 units right and 3 units up from origin?
  6. A point lies in the fourth quadrant and is 2 units below x-axis and 4 units right of origin. Write the coordinates.
  1. Explain in detail the signs of coordinates in all four quadrants with a labeled diagram.
  2. Plot and label the points: (3, 4), (-3, 4), (-3, -4), (3, -4) and mention their quadrants.
  3. Explain the method to locate a point in the Cartesian plane. Give at least two examples with diagrams.
  4. If a point P lies 5 units on the right of origin and 7 units below x-axis, determine its coordinates. Also mention the quadrant in which it lies.

Case Study:
A treasure map is based on a coordinate grid. The origin (0, 0) marks the beginning of a journey. According to the map, the explorer must go to point A located 6 units to the right and 4 units up from the origin. Then from A, move to point B which is 8 units left and 3 units down. After reaching B, the final destination is point C, which lies 4 units to the right and 7 units up. All positions are recorded using Cartesian coordinates.

  1. What are the coordinates of point A?
    • A. (6, 4)
    • B. (-6, -4)
    • C. (4, 6)
    • D. (0, 10)
  2. From point A to point B, how much total movement occurs horizontally?
    • A. 14 units right
    • B. 14 units left
    • C. 8 units left
    • D. 2 units right
  3. What are the coordinates of point B?
    • A. (-2, 1)
    • B. (2, 1)
    • C. (0, 1)
    • D. (-2, -1)
  4. What is the final coordinate of point C?
    • A. (2, 8)
    • B. (2, 9)
    • C. (4, 8)
    • D. (4, 9)
  5. In which quadrant does point C lie?
    • A. I
    • B. II
    • C. III
    • D. IV

Section A:
1-B, 2-A, 3-D, 4-A, 5-B, 6-D, 7-B, 8-B

Section B:
[Students' own brief answers]

Section C:
[Students' own long-form answers with diagrams]

Section D:
1-A, 2-C, 3-A, 4-B, 5-A

Our Free Online Tests on Mathematics & Other Subjects

IELTS Class 9 Class 10 Class 11 Class 12 R Programming LaTeX JEE