3.1.1 Basic Coordinate Geometry

Basic coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. It allows us to describe geometric shapes, such as lines, circles, and polygons, algebraically using equations. Here are some fundamental concepts and techniques:

1. The Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is a two-dimensional plane defined by two perpendicular axes: the horizontal $\ x$-axis and the vertical $\ y$-axis. The point where these axes intersect is called the origin, denoted as $\ (0, 0)$.

Each point on the plane is represented by an ordered pair $\ (x, y)$, where:

• $\ x$ is the horizontal distance from the origin (positive to the right, negative to the left).
• $\ y$ is the vertical distance from the origin (positive upward, negative downward).

2. Distance Between Two Points

The distance between two points $\ A(x_1, y_1)$ and $\ B(x_2, y_2)$ on the coordinate plane is given by the distance formula:

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

3. Midpoint of a Line Segment

The midpoint of a line segment connecting two points $\ A(x_1, y_1)$ and $\ B(x_2, y_2)$ is the point exactly halfway between them. It is given by the midpoint formula:

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

4. Slope of a Line

The slope of a line is a measure of its steepness and is defined as the ratio of the change in the$\ y$-coordinate to the change in the $\ x$-coordinate between two points on the line. If $\ A(x_1, y_1)$and $\ B(x_2, y_2)$ are two points on the line, the slope$\ m$ is:

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

• A positive slope indicates the line rises as it moves from left to right.
• A negative slope indicates the line falls as it moves from left to right.
• A slope of $0$ indicates a horizontal line.
• An undefined slope (division by $0$) indicates a vertical line.

5. Equation of a Line

The equation of a line can be expressed in several forms, with the most common being the slope-intercept form and the point-slope form.

Slope-Intercept Form

The slope-intercept form of the equation of a line is:

$y = mx + c$

• $\ m$ is the slope of the line.
• $\ c$ is the$\ y$-intercept (the point where the line crosses the $\ y -$axis).

Point-Slope Form

The point-slope form of the equation of a line passing through a point $\ (x_1, y_1)$ with slope $\ m$is:

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

6. Parallel and Perpendicular Lines

• Parallel Lines: Two lines are parallel if they have the same slope, i.e., $\ m_1 = m_2 .$
• Perpendicular Lines: Two lines are perpendicular if the product of their slopes is $\ -1$, i.e., $\ m_1 \times m_2 = -1 .$

Practice Problem: