Lines and slopes

Lines

Definitions

Line

A perfectly straight path that extends infinitely in both directions

Line segment

A line with endpoints (a defined start and end point)

This model shows the difference between a line and a line segment. A line continues forever in both directions, while a line segment has fixed endpoints. This visual helps you see how one extends infinitely and the other does not.

Real world example

A road that continues in both directions can be modeled as a line, while a fence between two posts is a line segment because it has endpoints.

Slopes

Definitions

Slope

A measure of both the steepness and direction of a line, calculated as the vertical change divided by the horizontal change ($\frac{rise}{run}$)

Slope tells you how a line moves as you go from left to right. It shows both how steep the line is and which direction it goes.

A helpful way to think about slope is “rise over run”:

• Rise = How much the line goes up or down
• Run = How much the line goes left or right

This model shows how slope works: The sign of the slope tells you which way the line moves as you go from left to right:

• A line with a positive slope goes up to the right.
• A line with a negative slope goes down to the right.

Sidenote

Remember this!

A perfectly horizontal line has a slope of $0$ because it has no rise. Its rise is $0$, so its slope is always $\frac{0}{a}$. No matter how large the run is, $0$ divided by any number is $0$.

A perfectly vertical line has an undefined slope because it has no run. Its run is $0$, so its slope is always a $\frac{number}{0}$. Any number divided by $0$ is undefined.

Types of slope

Lines can have different types of slope depending on how they move from left to right.

Positive slope: The line goes up as you move to the right
Negative slope: The line goes down as you move to the right
Zero slope: The line is horizontal (no rise)
Undefined slope: The line is vertical (no run)

Knowledge check: A line goes down as you move from left to right. What type of slope does it have?

(spoiler)

A negative slope.

Interactions of lines

Definitions

Parallel lines

Lines that have the same slope but different starting points. They never intersect and remain a constant distance apart.

When comparing two lines, it’s helpful to look at their slopes. The slope tells you whether lines will meet, stay apart, or overlap.

Two lines can interact in one of three ways:

1. They can intersect at exactly one point
2. They can never intersect
3. They can overlap and intersect at infinite points.

• Lines intersect when they have different slope values. Since they rise or fall at different rates, they will meet at some point.
• Lines never intersect when they have the same slope value but different starting points. Since they travel at the same rate but start in different locations, they stay a constant distance apart. These are also called parallel lines.
• Lines intersect infinitely when they have the same slope and the same starting point. Because they begin at the same location and travel at the exact same rate, they are the same line.

Perpendicular lines

Definitions

Perpendicular lines

Lines that meet at exactly one point and form four $90$° angles (right angles). Their slopes are opposite reciprocals of each other.

One important type of intersecting lines is perpendicular lines. They meet at exactly one point and form four $90^{\circ }$ angles (right angles).

You can also recognize perpendicular lines by looking at their slopes. The slopes of perpendicular lines are opposite reciprocals.

That means:

• The signs are opposite, and
• The fractions are flipped (reciprocal).

Think: flip the fraction and change the sign.

So if one line has a slope of $+\frac{2}{3}$, the line perpendicular to it has a slope of $-\frac{3}{2}$.

In the graph above, the two lines are perpendicular because their slopes are opposite reciprocals.

Practice problem

Example:

If one line has a slope of $\frac{3}{2}$, what is the slope of a perpendicular line?

To solve, flip the fraction and change the sign:

$$-\frac{2}{3}$$

Knowledge check: A line has a slope of $-5$. What is the slope of a line perpendicular to it?

(spoiler)

Rewrite $-5$ as $-\frac{5}{1}$, then flip the fraction and change the sign:

$$\frac{1}{5}$$