Geometry (A): READ: Slopes Reading

Slopes of Lines

Learning Objectives

Identify and compute slope in the coordinate plane.
Use the relationship between slopes of parallel lines.
Use the relationship between slopes of perpendicular lines.
Plot a line on a coordinate plane using different methods.

Introduction

You may recall from algebra that you spent a lot of time graphing lines in the $xy-$ coordinate plane. How are those lines related to the lines we’ve studied in geometry? Lines on a graph can be studied for their slope (or rate of change), and how they intersect the $x-$ and $y-$ axes.

Slope in the Coordinate Plane

If you look at a graph of a line, you can think of the slope as the steepness of the line (assuming that the $x-$ and $y-$ scales are equal. Mathematically, you can calculate the slope using two different points on a line. Given two points $(x_1, y_1)$ and $(x_2, y_2)$ the slope is computed as:

$\text{slope} = \frac{(y_2 - y_1)} {(x_2 - x_1)}$

You may have also learned this as “slope equals rise over run.” In other words, first calculate the distance that the line travels up (or down), and then divide that value by the distance the line travels left to right. The left to right distance in this scenario is referred to as the run.

A line that goes up from left to right has positive slope, and a line that goes down from left to right has negative slope.

Example 1

What is the slope of a line that travels through the points $(2, 2)$ and $(4, 6)$ ?

You can use the previous formula to find the slope of this line. Let’s say that $(x_1, y_1)$ is $(2, 2)$ and $(x_2, y_2)$ is $(4, 6)$ . Then we find the slope as follows:

$\text{slope} & = \frac{(y_2 - y_1)} {(x_2 - x_1)}\\ \text{slope} & = \frac{(6 - 2)} {(4 - 2)}\\ \text{slope} & = \frac{4} {2}\\ \text{slope} & = 2$

The slope of the line in Example 1 is 2. Let’s look at what that means graphically.

These are the two points in question. You can see that the line rises $4 \;\mathrm{units}$ as it travels $2\;\mathrm{units}$ to the right. So, the rise is $4\;\mathrm{units}$ and the run is $2\;\mathrm{units}$ . Since $4 \div 2 = 2$ , the slope of this line is $2$ .

Notice that the slope of the line in example 1 was 2, a positive number. Any line with a positive slope will travel up from left to right. Any line with a negative slope will travel down from left to right. Check this fact in example 2.

Example 2

What is the slope of the line that travels through $(1,9)$ and $(3,3)$ ?

Use the formula again to identify the slope of this line.

$\text{slope} & = \frac{(y_2 - y_1)} {(x_2 - x_1)}\\ \text{slope} & = \frac{(3 - 9)} {(3 - 1)}\\ \text{slope} & = \frac{-6} {2}\\ \text{slope} & = -3$

The slope of this line in Example 2 is $-3.$ It will travel down to the right. The points and the line that connects them is shown below.

There are other types of lines with their own distinct slopes. Perform these calculations carefully to identify their slopes.

Example 3

What is the slope of a line that travels through $(4,4)$ and $(8,4)$ ?

Use the formula to find the slope of this line.

$\text{slope} & = \frac{(y_2 - y_1)} {(x_2 - x_1)}\\ \text{slope} & = \frac{(4 - 4)} {(8 - 4)}\\ \text{slope} & = \frac{0} {4}\\ \text{slope} & = 0$

This line, which is horizontal, has a slope of $0.$ Any horizontal line will have a slope of $0.$

Example 4

What is the slope of a line through $(3,2)$ and $(3,6)$ ?

Use the formula to identify the slope of this line.

$\text{slope} & = \frac{(y_2 - y_1)} {(x_2 - x_1)}\\ \text{slope} & = \frac{(6 - 2)} {(3 - 3)}\\ \text{slope} & = \frac{4} {0}\\ \text{slope} & = \text{undefined}$

The line in this example is vertical and we found that the numerical value of the slope is undefined.

In review, if you scan a graph of a line from left to right, then,

Lines with positive slopes point up to the right,
Lines with negative slopes point down to the right,
Horizontal lines have a slope of zero, and
Vertical lines have undefined slope. You can use these general rules to check your work when working with slopes and lines.

Slopes of Parallel Lines

Now that you know how to find the slope of lines using coordinates, you can think about how lines and their slopes are related.

Slope of Parallel Lines Theorem

If two lines in the coordinate plane are parallel they will have the same slope, conversely, if two lines in the coordinate plane have the same slope, those lines are parallel.

Note the proof of this theorem will have to wait until you have more mathematical tools, but for now you can use it to solve problems.

Example 5

Which of the following could represent the slope of a line parallel to the one following?

: D. $1$
: C. $\frac{1} {4}$
: B. $-1$
: A. $-4$

Since you are looking for the slope of a parallel line, it will have the same slope as the line in the diagram. First identify the slope of the line given, and select the answer with that slope. You can use the slope formula to find its value. Pick two points on the line. For example, $(-1,5)$ and $(3,1)$ .

$\text{slope} & = \frac{(y_2 - y_1)} {(x_2 - x_1)}\\ \text{slope} & = \frac{(1 - 5)} {(3 - (-1))}\\ \text{slope} & = \frac{-4} {4}\\ \text{slope} & = -1$

The slope of the line in the diagram is $-1.$ The answer is B.

Slopes of Perpendicular Lines

Parallel lines have the same slope. There is also a mathematical relationship for the slopes of perpendicular lines.

Perpendicular Line Slope Theorem

The slopes of perpendicular lines will be the opposite reciprocal of each other.

Another way to say this theorem is, if the slopes of two lines multiply to $-1,$ then the two lines are perpendicular.

The opposite reciprocal can be found in two steps. First, find the reciprocal of the given slope. If the slope is a fraction, you can simply switch the numbers in the numerator and denominator. If the value is not a fraction, you can make it into one by putting a $1$ in the numerator and the given value in the denominator. The reciprocal of $\frac{2} {3}$ is $\frac{3} {2}$ and the reciprocal of $5$ is $\frac{1} {5}$ . The second step is to find the opposite of the given number. If the value is positive, make it negative. If the value is negative, make it positive. The opposite reciprocal of $\frac{2} {3}$ is $- \frac{3} {2}$ and the opposite reciprocal of $5$ is $- \frac{1} {5}$ .

Example 6

Which of the following could represent the slope of a line perpendicular to the one shown below?

: D. $\frac{7} {5}$
: C. $\frac{5} {7}$
: B. $- \frac{5} {7}$
: A. $- \frac{7} {5}$

Since you are looking for the slope of a perpendicular line, it will be the opposite reciprocal of the slope of the line in the diagram. First identify the slope of the line given, then find the opposite reciprocal, and finally select the answer with that value. You can use the slope formula to find the original slope. Pick two points on the line. For example, $(-3,-2)$ and $(4,3)$ .

$\text{slope} & = \frac{(y_2 - y_1)} {(x_2 - x_1)}\\ \text{slope} & = \frac{(3 - (-2))} {(4 - (-3))}\\ \text{slope} & = \frac{(3 + 2)} {(4 + 3)}\\ \text{slope} & = \frac{5} {7}$

The slope of the line in the diagram is $\frac{5} {7}$ . Now find the opposite reciprocal of that value. First swap the numerator and denominator in the fraction, then find its opposite. The opposite reciprocal of $\frac{5} {7}$ is $- \frac{7} {5}$ . The answer is A.

Graphing Strategies

There are a number of ways to graph lines using slopes and points. This is an important skill to use throughout algebra and geometry. If you write an equation in algebra, it can help you to see the general slope of a line and understand its trend. This could be particularly helpful if you are making a financial analysis of a business plan, or are trying to figure out how long it will take you save enough money to buy something special. In geometry, knowing the behavior of different types of functions can be helpful to understand and make predictions about shapes, sizes, and trends.

There are two simple ways to create a linear graph. The first is to use two points that are given to you. Plot them on a coordinate grid, and draw a line segment connecting them. This segment can be expanded to represent the entire line that passes through those two points.

Example 7

Draw the line that passes through $(-3, 3)$ and $(4, -2).$

Begin by plotting these points on a coordinate grid. Remember that the first number in the ordered pair represents the $x-$ value and the second number represents the $y-$ value.

Draw a segment connecting these two points and extend that segment in both directions, adding arrows to both ends. This shows the only line that passes through points $(-3,3)$ and $(4,-2).$

The other way to graph a line is using one point and the slope. Start by plotting the given point and using the slope to calculate another point. Then you can draw the segment and extend it as you did in the previous example.

Example 8

Draw the line that passes through $(0,1)$ and has a slope of $3.$

Begin by plotting the given point on a coordinate grid.

If the slope is $3$ , you can interpret that as $\frac{3} {1}$ . The fractional expression makes it easier to identify the rise and the run. So, the rise is $3$ and the run is $1$ . Find and plot a point that leaves the given coordinate and travels up three units and one unit to the right. This point will also be on the line.

Now you have plotted a second point on the line at $(1,4).$ You can connect these two points, extend the segment, and add arrows to show the line that passes through $(0,1)$ with a slope of $3.$

Lesson Summary

In this lesson, we explored how to work with lines in the coordinate plane. Specifically, we have learned:

How to identify slope in the coordinate plane.
How to identify the relationship between slopes of parallel lines.
How to identify the relationship between slopes of perpendicular lines.
How to plot a line on a coordinate plane using different methods.

These skills will help you solve many different types of problems. Always be on the lookout for new and interesting ways to apply concepts of slope, parallel and perpendicular lines, and graphing to mathematical situations.

Points to Consider

Now that you have studied slope, graphing techniques, and other issues related to lines, you can learn about their algebraic properties. In the next lesson, you’ll learn how to write different types of equations that represent lines in the coordinate plane.