3.1-Introduction

3.1-Introduction Important Formulae

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Grade 9 → Math → Coordinate Geometry → 3.1-Introduction

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

  • Get familiarized with the concepts of coordinate geometry such as origin, x and y axes, intercepts.

Coordinate Geometry, also known as Analytical Geometry, is the study of geometric figures using a coordinate system. In Grade 9, students learn to represent points, lines, and shapes in a plane using ordered pairs.

The Cartesian coordinate system is a fundamental aspect of coordinate geometry. It consists of two perpendicular axes:

  • X-axis: The horizontal axis.
  • Y-axis: The vertical axis.

The point where the X-axis and Y-axis intersect is called the origin, denoted as (0, 0). Each point in the plane can be represented by an ordered pair $(x, y)$, where:

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

The coordinate plane is divided into four quadrants:

  1. Quadrant I: Both $x$ and $y$ are positive $(+,+)$.
  2. Quadrant II: $x$ is negative and $y$ is positive $(-,+)$.
  3. Quadrant III: Both $x$ and $y$ are negative $(-,-)$.
  4. Quadrant IV: $x$ is positive and $y$ is negative $(+,-)$.

To find the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ in the coordinate plane, the formula is:

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

Another important concept is the midpoint of a line segment connecting two points. The coordinates of the midpoint $M$ between points $(x_1, y_1)$ and $(x_2, y_2)$ are given by:

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

In addition to distance and midpoint, students also explore the slope of a line. The slope $m$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is calculated as:

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

The slope indicates the steepness and direction of a line. A positive slope means the line rises, while a negative slope indicates it falls. A slope of zero represents a horizontal line, and an undefined slope represents a vertical line.

The equation of a straight line can be expressed in various forms, with the slope-intercept form being one of the most common:

$y = mx + c$

Where $m$ is the slope and $c$ is the y-intercept, the point where the line crosses the y-axis.

Another form is the point-slope form, which is useful when the slope and a point on the line are known:

$y - y_1 = m(x - x_1)$


www.cuemath.com, CC BY-SA 4.0, via Wikimedia Commons

How will you describe the position of a table lamp on your study table to another person?

Solution:
Consider the lamp as a point and table as a plane. Choose any two perpendicular edges of the table. Measure the distance of the lamp from the longer edge, suppose it is 25 cm. Again, measure the distance of the lamp from the shorter edge, and suppose it is 30 cm. You can write the position of the lamp as (30, 25) or (25, 30), depending on the order you fix.

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(Street Plan) : A city has two main roads which cross each other at the centre of the city. These two roads are along the North-South direction and East-West direction. All the other streets of the city run parallel to these roads and are 200 m apart. There are 5 streets in each direction. Using 1cm = 200 m, draw a model of the city on your notebook. Represent the roads/streets by single lines.
There are many cross- streets in your model. A particular cross-street is made by two streets, one running in the North - South direction and another in the East - West direction. Each cross street is referred to in the following manner : If the 2nd street running in the North - South direction and 5th in the East - West direction meet at some crossing, then we will call this cross-street (2, 5). Using this convention, find:

(i)  How many cross - streets can be referred to as (4, 3).
(ii)  How many cross - streets can be referred to as (3, 4).

Solution:
Both the cross-streets are marked in the figure above. They are uniquely found because of the two reference lines we have used for locating them.

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