Cartesian Plane/Coordinates

What is the Cartesian Plane?

The Cartesian plane is a two-dimensional plane used for plotting points, lines, and shapes. It is defined by two perpendicular axes:

• $x-axis$ (horizontal)
• $y-axis$ (vertical) These axes intersect at the origin, denoted as $(0, 0)$.

The plane is divided into four quadrants:

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

Coordinates of a Point

A point on the Cartesian plane is represented by an ordered pair ($x, y$):

• $x$: The horizontal distance from the origin.
• $y$: The vertical distance from the origin. For example, the point $(3, -2)$ is located 3 units to the right of the origin and 2 units down.

Distance Formula

The distance $d$ between two points ($x_1, y_1$) and ($x_2, y_2$) is given by:

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

Midpoint Formula

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

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

Equation of a Line

A straight line on the Cartesian plane can be represented by its equation in the form:

$$y = mx + c$$

where:

• $m$ is the slope of the line, calculated as $\frac{y_2 - y_1}{x_2 - x_1}$
• $c$ is the $y-intercept$, the point where the line crosses the $y-axis$.

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Worked Examples

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Summary

• The Cartesian plane is defined by two perpendicular axes ($x$ and $y$) intersecting at the origin $(0, 0)$
• Points are represented by ordered pairs $(x, y)$
• Key formulas:
  • Distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
  • Midpoint formula: $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$
  • Line equation: $y = mx + c$, where $m = \frac{y_2 - y_1}{x_2 - x_1}$
• Practice applying these concepts to strengthen understanding.