Class 9 — Mathematics (NCERT)

Coordinate Geometry: The Cartesian Plane and Plotting Points

Class 9

✓By the end of this lesson students will be able to understand the concept of a Cartesian plane.
✓By the end of this lesson students will be able to identify the x-axis, y-axis, origin, and quadrants.
✓By the end of this lesson students will be able to determine the coordinates (abscissa and ordinate) of a given point.
✓By the end of this lesson students will be able to plot a point on the Cartesian plane given its coordinates.
✓By the end of this lesson students will be able to locate points on the axes and in different quadrants.

Key concepts

The Cartesian Plane

The Cartesian plane, also known as the coordinate plane, is a system used to locate points in a two-dimensional space. It consists of two mutually perpendicular number lines: a horizontal line called the X-axis (denoted as X'OX) and a vertical line called the Y-axis (denoted as Y'OY). The point where these two axes intersect is called the origin, denoted by 'O'. The axes divide the plane into four regions called quadrants. The regions are numbered I, II, III, and IV in an anti-clockwise direction starting from the upper right region.

Coordinates of a Point

Every point in the Cartesian plane can be uniquely identified by an ordered pair of numbers, called its coordinates. The first number in the pair is the x-coordinate, also known as the abscissa. It represents the perpendicular distance of the point from the Y-axis. The second number is the y-coordinate, also known as the ordinate. It represents the perpendicular distance of the point from the X-axis. The coordinates are always written in the form (x, y), where 'x' is the abscissa and 'y' is the ordinate. For example, the coordinates of the origin are (0, 0).

(x, y)

Plotting a Point

To plot a point (x, y) on the Cartesian plane, we follow these steps: \n1. Start from the origin (0, 0).\n2. Move 'x' units horizontally along the X-axis. If 'x' is positive, move to the right; if 'x' is negative, move to the left. If 'x' is zero, stay on the Y-axis.\n3. From the position reached in step 2, move 'y' units vertically parallel to the Y-axis. If 'y' is positive, move upwards; if 'y' is negative, move downwards. If 'y' is zero, stay on the X-axis.\n4. Mark the final position and label it with the given point's name and its coordinates.

Key facts to remember

1The x-axis and y-axis are perpendicular to each other and intersect at the origin (0, 0).
2The x-coordinate is called the abscissa, and the y-coordinate is called the ordinate.
3The coordinates of a point are always written as an ordered pair (abscissa, ordinate) or (x, y).
4Points on the X-axis have their y-coordinate as 0, i.e., they are of the form (x, 0).
5Points on the Y-axis have their x-coordinate as 0, i.e., they are of the form (0, y).
6The signs of coordinates in the four quadrants are: Quadrant I (+, +), Quadrant II (-, +), Quadrant III (-, -), Quadrant IV (+, -).

Worked examples

Example 1

Plot the following points on a Cartesian plane: A(3, 4), B(-2, 3), C(-4, -2), D(5, -1).

IDraw the X-axis and Y-axis, marking the origin O(0,0). Label the positive and negative directions on both axes and choose a suitable scale (e.g., 1 unit = 1 cm).

IITo plot A(3, 4): Start at the origin. Move 3 units to the right along the X-axis. From there, move 4 units upwards parallel to the Y-axis. Mark this point as A.

IIITo plot B(-2, 3): Start at the origin. Move 2 units to the left along the X-axis. From there, move 3 units upwards parallel to the Y-axis. Mark this point as B.

IVTo plot C(-4, -2): Start at the origin. Move 4 units to the left along the X-axis. From there, move 2 units downwards parallel to the Y-axis. Mark this point as C.

VTo plot D(5, -1): Start at the origin. Move 5 units to the right along the X-axis. From there, move 1 unit downwards parallel to the Y-axis. Mark this point as D.

Answer

Point A(3, 4) is in Quadrant I.\nPoint B(-2, 3) is in Quadrant II.\nPoint C(-4, -2) is in Quadrant III.\nPoint D(5, -1) is in Quadrant IV.

Ensure your axes are clearly labelled and the scale is consistent.

Example 2

Write the coordinates of the points P, Q, R, S if P is 3 units to the right of the origin and 2 units above the X-axis, Q is 4 units to the left of the origin and 1 unit above the X-axis, R is 2 units to the left of the origin and 3 units below the X-axis, and S is 5 units to the right of the origin and on the X-axis.

IFor point P: '3 units to the right of the origin' means x = +3. '2 units above the X-axis' means y = +2. Hence, the coordinates of P are (3, 2).

IIFor point Q: '4 units to the left of the origin' means x = -4. '1 unit above the X-axis' means y = +1. Hence, the coordinates of Q are (-4, 1).

IIIFor point R: '2 units to the left of the origin' means x = -2. '3 units below the X-axis' means y = -3. Hence, the coordinates of R are (-2, -3).

IVFor point S: '5 units to the right of the origin' means x = +5. 'on the X-axis' means y = 0. Hence, the coordinates of S are (5, 0).

Answer

P(3, 2), Q(-4, 1), R(-2, -3), S(5, 0).

Remember that points on the X-axis have a y-coordinate of 0, and points on the Y-axis have an x-coordinate of 0.

Example 3

Identify the quadrant or axis on which each of the following points lies: (2, 5), (-3, 1), (0, -4), (6, 0), (-1, -5).

IFor (2, 5): The x-coordinate (2) is positive and the y-coordinate (5) is positive. Therefore, the point lies in Quadrant I.

IIFor (-3, 1): The x-coordinate (-3) is negative and the y-coordinate (1) is positive. Therefore, the point lies in Quadrant II.

IIIFor (0, -4): The x-coordinate (0) is zero. Therefore, the point lies on the Y-axis (specifically, on the negative Y-axis).

IVFor (6, 0): The y-coordinate (0) is zero. Therefore, the point lies on the X-axis (specifically, on the positive X-axis).

VFor (-1, -5): The x-coordinate (-1) is negative and the y-coordinate (-5) is negative. Therefore, the point lies in Quadrant III.

Answer

(2, 5) is in Quadrant I.\n(-3, 1) is in Quadrant II.\n(0, -4) is on the Y-axis.\n(6, 0) is on the X-axis.\n(-1, -5) is in Quadrant III.

A point with an x-coordinate of zero lies on the Y-axis. A point with a y-coordinate of zero lies on the X-axis.

Common mistakes

✗Confusing the x-coordinate (abscissa) with the y-coordinate (ordinate) and plotting (y, x) instead of (x, y).
✗Incorrectly identifying positive and negative directions on the axes (e.g., moving left for positive x).
✗Not starting from the origin (0, 0) when plotting a point.
✗Miscounting units on the graph paper, leading to inaccurate plotting.
✗Forgetting that points on the axes have one of their coordinates as zero.

Exam tips

★Always draw clear and properly labelled axes (X, X', Y, Y', O) with arrows indicating positive directions.
★Choose an appropriate scale for your axes to fit all points on the graph paper and label the scale clearly.
★Mark the points distinctly on the graph and label them with their names and coordinates (e.g., P(3, 2)).
★Double-check the signs of the coordinates before plotting to ensure the point is placed in the correct quadrant or on the correct axis.

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