This section of our PERT Study Guide covers coordinate planes. You will learn how to interact with coordinates, how to plot lines and equations on a plane, and how to use coordinate geometry to form an equation or define a point.
Slope of a Line
Slope, denoted as $m$, measures the steepness and direction of a line. It shows how much the $y$-value (vertical change) changes for each unit increase in the $x$-value (horizontal change).
For any two points on a line, ($x_1, y_1$) and ($x_2, y_2$), the slope is calculated as:
$m = \dfrac{y_2 − y_1}{x_2 − x_1}$
You can remember this as rise over run; the $y$-value is the change in “rise” whereas the $x$-value is the change in a horizontal “run.”
The value of the slope, $m$, will tell you how the line looks on a graph.
- Positive Slope ($m > 0$): The line rises from left to right.
- Negative Slope ($m < 0$): The line falls from left to right.
- Zero Slope ($m = 0$): Horizontal line, with no vertical change as $x$ changes.
- Undefined Slope: Vertical line (occurs when the formula for $m$ results in dividing by $0$).
Example: To find the slope of a line passing through $(3, 4)$ and $(7, 10)$, calculate:
$m = \dfrac{(10 − 4)}{(7 − 3)} = 1.5$
This slope of $1.5$ indicates a rise by $1.5$ units for each unit increase in $x$.
Distance Formula
The distance formula calculates the straight-line distance between two points on a coordinate plane.
For two points ($x_1, y_1$) and ($x_2, y_2$), the distance $d$ is given by:
$d= \sqrt{(x_2 − x_1)^2+(y_2 − y_1)^2}$
Example: To find the distance between $(1, 2)$ and $(5, 6)$, substitute into the formula:
$d= \sqrt{(5 − 1)^2+(6 − 2)^2} $ $ = 4 \sqrt{2}$
Equation of a Line
Lines can be defined based on an equation. There are a few common forms in which this equation can be written.
Slope-Intercept Form: $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept (where the line comes in contact with the $y$-axis).
You can find this equation if you know the slope and one coordinate. Substitute the slope for $m$ and the coordinates for $y$ and $x$ into $y = mx + b$, and then solve for $b$.
Example: If a line has a slope of $2$ and the point $(1, 3)$, then its equation in slope-intercept form would be $y = 2x + 1$.
Other Forms:
- Point-Slope Form: $y − y_1 = m(x − x_1)$
- Standard Form: $Ax + By = C$, with integers for $A$, $B$, and $C$.
Perpendicular and Parallel Lines
Parallel lines are lines that never touch. They share the same slope but different intercepts.
Perpendicular lines cross at a right angle. The product of the slopes of two perpendicular lines is $−1$; if one line has slope $m$, a perpendicular line has slope $\frac{−1}{m}$.
Example: A line with the slope $\frac{2}{3}$ is perpendicular to a line with the slope $−\frac{3}{2}$.
Key Tip: Make sure you can calculate slope and distance. Practice using the formulas to reinforce learning.