3.2-Cartesian System

3.2-Cartesian System Important Formulae

You are currently studying
Grade 9 → Math → Coordinate Geometry → 3.2-Cartesian System

After successful completion of this topic, you should be able to:

  • To be able to state coordinates of a point on a 2-D plane.

The Cartesian system, developed by René Descartes, is a fundamental concept in coordinate geometry that allows us to represent geometric shapes using ordered pairs of numbers. This system forms the basis for graphing linear equations and understanding the relationship between algebra and geometry.

In a Cartesian coordinate system, we use two perpendicular axes: the horizontal axis (x-axis) and the vertical axis (y-axis). These axes intersect at a point called the origin, which has coordinates $(0, 0)$. The axes divide the plane into four quadrants:

  • Quadrant I: Both $x$ and $y$ are positive $(+,+)$
  • Quadrant II: $x$ is negative and $y$ is positive $(-,+)$
  • Quadrant III: Both $x$ and $y$ are negative $(-,-)$
  • Quadrant IV: $x$ is positive and $y$ is negative $(+,-)$

The position of any point in the Cartesian plane can be defined by an ordered pair $(x, y)$, where:

  • $x$ is the horizontal distance from the origin.
  • $y$ is the vertical distance from the origin.

To plot a point, start at the origin. Move $x$ units along the x-axis (right for positive, left for negative) and then move $y$ units parallel to the y-axis (up for positive, down for negative).

For example, to plot the point $(3, 2)$:

  1. Start at the origin $(0, 0)$.
  2. Move 3 units to the right along the x-axis.
  3. From there, move 2 units up parallel to the y-axis.

In addition to plotting points, the Cartesian system allows for the representation of lines and curves. The general equation of a line in slope-intercept form is given by:

$$y = mx + b$$

Where:

  • $m$ is the slope of the line, which indicates its steepness and direction.
  • $b$ is the y-intercept, the point where the line crosses the y-axis.

The slope $m$ can be calculated using the formula:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

Where $(x_1, y_1)$ and $(x_2, y_2)$ are two distinct points on the line. A positive slope indicates that the line rises as you move from left to right, while a negative slope indicates that the line falls.

The Cartesian system is also used for distance and midpoint calculations. The distance $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ can be calculated using the distance formula:

$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

The midpoint $M$ of the line segment joining the points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

$$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$

Understanding the Cartesian system is crucial for solving problems in coordinate geometry, as it lays the foundation for more advanced mathematical concepts.


A small portion of the Cartesian coordinate system, showing the origin, axes, and the four quadrants, with illustrative points and grid.
Kbolino, Public domain, via Wikimedia Commons

Write the answer of each of the following questions:
(i)  What is the name of horizontal and the vertical lines drawn to determine the position of any point in the Cartesian plane?
(ii)  What is the name of each part of the plane formed by these two lines? 

(iii) Write the name of the point where these two lines intersect.

Solution:
(i) The x - axis and the y - axis.
(ii) Quadrants.
(iii) The origin.

Suggest improvements/Report errors

See Fig. 3.14, and write the following: 

(i)  The coordinates of B.
(ii)  The coordinates of C. 

(iii) The point identified by the coordinates(–3,–5).

iv) The point identified by the coordinates (2, – 4).

(v) The abscissa of the point D.

(vi) The ordinate of the point H.

(vii) The coordinates of the point L.

(viii) The coordinates of the point M.

Solution:
(i) (–5,2)
(ii)(5,–5)
(iii)E
(iv)G
(v)6
(vi)–3
(vii) (0,5)
(viii) (–3,0)

Suggest improvements/Report errors