Class 9 - Cartesian System

The Cartesian system is used to locate the position of an object on a plane using two perpendicular lines. This study was initially developed by the French philosopher and mathematician René Déscartes in the seventh century. This concept is vastly applied in real-life to locate coordinates of any objects on maps. Let’s understand this concept of representing geometrical figures in mathematics using a cartesian coordinate system along with its graphical representation and examples.

Table of Contents

• What is Cartesian Plane
• Abscissa and Ordinates
• What are Quadrants
• Solved Examples on Cartesian System

What is Cartesian Plane

Understanding the cartesian plane also known as a coordinate plane is important for cartesian coordinate geometry. It can be defined as a plane formed by intersection of two perpendicular lines called the axes of a plane. The horizontal axis is called the x-axis and the vertical axis is called the y-axis. The point at which both these lines intersect is called the origin ‘O’ represented through coordinates O(0,0). Both the X and Y axis extend in positive and negative directions starting from the origin. The positive numbers lie in the directions OX and OY. So they are called the positive directions of the x-axis and the y-axis, respectively. The negative numbers lie in the direction OX′ and OY′. So OX′ and OY′ are called the negative directions of the x - axis and the y - axis, respectively. 

Let’s understand basics terms related in a Cartesian plane one by one here:

• X-axis: The horizontal number line. Positive values lie to the right of the origin; negative values to the left.

• Y-axis: The vertical number line. Positive values lie above the origin; negative values below.

• Origin: The point of intersection of the x-axis and y-axis, denoted by O (0, 0).

• Ordered pair (x, y): The coordinates of any point, where x is the abscissa (horizontal distance) and y is the ordinate (vertical distance).

Any point of a cartesian plane is represented as P(x,y), where x is the distance from the y-axis and y is the distance from x-axis.

Abscissa and Ordinates

Each point is represented on a Cartesian plane through x and y coordinates. The x - coordinate of a point is its perpendicular distance from the y - axis measured along the x -axis (positive along the positive direction of the x - axis and negative along the negative direction of the x - axis).

The x-coordinate of a point is called the abscissa and the y-coordinate of a point  is called the ordinate. Let’s consider P and Q are two points on a cartesian plane with coordinates P(3, 4) and Q(-6, -2) respectively then, x1 = 3 and y1  = 4 and x2 = -6 and y2 = -2 are called coordinates of P and Q respective that can be represented graphically on a coordinate plane. The x-coordinates of the point P(x1 = 3) and Q(x2 = -6) are called the abscissa and y1  = 4 and y2 = -2 are called the ordinates.

What are Quadrants

The cartesian plane consists of two axes that divides the plane into four equal parts called the four quadrants of a cartesian plane. A quadrant is the area enclosed by the intersection of the x and y axis. Each point on the quadrants is represented through its x and y coordinates as: P(x, y), where x is the distance from the y-axis and y is the distance from x-axis. These two values are called the coordinates of a point. The value of each coordinate can be positive or negative based on the quadrant it is located in. The relationship between the signs of the coordinates of a point and the quadrant of a point in which it lies are:

• If a point is in the 1st quadrant, then the point will be in the form (+, +), since the 1st quadrant is enclosed by the positive x - axis and the positive y - axis.

• If a point is in the 2nd quadrant, then the point will be in the form (–, +), since the 2nd quadrant is enclosed by the negative x - axis and the positive y - axis.

• If a point is in the 3rd quadrant, then the point will be in the form (–, –), since the 3rd quadrant is enclosed by the negative x - axis and the negative y - axis.

• If a point is in the 4th quadrant, then the point will be in the form (+, –), since the 4th quadrant is enclosed by the positive x - axis and the negative y - axis.
  
  

Solved Examples on Cartesian System

Example 1: Find the abscissa and ordinate of a point R with coordinates as R(-3, 2).
Solution: The coordinates of the given point R is (-3, 2)
Abscissa: The x-coordinate of the given point R is called its abscissa = -3
Ordinate: The y-coordinate of the given point R is called its ordinate = 2

Example 2: If the x-coordinate of a point is -5 and y-coordinate is -6 then in which coordinate does the point lies?
Solution: Let's understand the coordinates of the given point:
Abscissa or the x-coordinate of the given point is -5 and the y-coordinate of the given point is -6.
So, the point is R = (-5, -6). Since both the values are negative the point is located in the third quadrant.

Example 3: Where do the following points lie?
(i) (0, −8) (ii) (7, 0) (iii) (0, 0)
Solution:

• (0, −8): x = 0, so the point lies on the y-axis (negative direction, below origin).
• (7, 0): y = 0, so the point lies on the x-axis (positive direction, right of origin).
• (0, 0): This is the origin — the intersection of both axes.