Lesson 3: Using Slope in Problems

Property

A line passing through two points $A$ and $B$ is denoted as $\overleftrightarrow{AB}$ or $\overleftrightarrow{BA}$. The arrow symbol indicates the line extends infinitely in both directions through the two points.

Examples

Property

To graph a line given a point and the slope:

1. Plot the given point.
2. Use the slope formula $m = \frac{\operatorname{rise}}{\operatorname{run}}$ to identify the rise and the run.
3. Starting at the given point, count out the rise and run to mark the second point.
4. Connect the points with a line.

Examples

• To graph a line through $(1, 2)$ with slope $m = \frac{3}{5}$, start at $(1, 2)$, move up 3 units and right 5 units to plot a second point at $(6, 5)$.
• To graph a line through $(-2, 4)$ with slope $m = -3$, treat the slope as $\frac{-3}{1}$. From $(-2, 4)$, move down 3 units and right 1 unit to plot a second point at $(-1, 1)$.
• To graph a line through the y-intercept $(0, -1)$ with slope $m = \frac{2}{3}$, start at $(0, -1)$, move up 2 units and right 3 units to plot a second point at $(3, 1)$.

Explanation

Think of it as 'point and directions'. Start by plotting the given point. Then use the slope's rise and run as steps to find a second point. Connect them to draw your line.

Property

The midpoint $M(\bar{x}, \bar{y})$ of the line segment joining the points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$ is found by averaging the coordinates:

Examples

• To find the midpoint of the segment joining $(2, 5)$ and $(8, 11)$, calculate $M = (\frac{2+8}{2}, \frac{5+11}{2}) = (5, 8)$.
• The midpoint of the segment between $(-3, 9)$ and $(5, -2)$ is $M = (\frac{2}{2}, \frac{7}{2}) = (1, 3.5)$.
• If a triangle has vertices at $(0,0)$ and $(2a, 0)$, the midpoint of that base is $(\frac{0+2a}{2}, \frac{0+0}{2}) = (a, 0)$.

Explanation

The Midpoint Formula finds the exact center of a line segment. It works by taking the mathematical average of the $x$-coordinates and the average of the $y$-coordinates to pinpoint the exact halfway mark. In coordinate proofs, this is the tool you use whenever a theorem mentions a "bisector" or a "median."