CK-12 Geometry
Page 11
Are lines and parallel in the following diagram? Justify your answer.
Are lines and parallel in the diagram below? Justify your answer.
For exercises 6-13, use the following diagram. Line and . Find each angle and give a justification for each of your answers.
________.
________.
________.
________.
________.
________.
________.
________.
Review Answers
Yes. If alternate interior angles are congruent, then the lines are parallel
No. Since alternate exterior angles are NOT congruent the lines are NOT parallel
No. Since alternate interior angles are NOT congruent, the lines are NOT parallel.
Yes. If corresponding angles are congruent, then the lines are parallel
Yes. If exterior angles on the same side of the transversal are supplementary, then the lines are parallel
. Since and are a linear pair, they are supplementary
. is an interior angle on the same side of transversal with the angle marked So and the angle are supplementary and
. is a vertical angle to the angle marked
. It is a corresponding angle with angle
. It is a linear pair with a right angle
. It is a corresponding angle with the angle marked
. It is a vertical angle with
. It is a linear pair with the angle marked
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 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 and 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 and scales are equal. Mathematically, you can calculate the slope using two different points on a line. Given two points and the slope is computed as:
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 and ?
You can use the previous formula to find the slope of this line. Let’s say that is and is . Then we find the slope as follows:
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 as it travels to the right. So, the rise is and the run is . Since , the slope of this line is .
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 and ?
Use the formula again to identify the slope of this line.
The slope of this line in Example 2 is 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 and ?
Use the formula to find the slope of this line.
This line, which is horizontal, has a slope of Any horizontal line will have a slope of
Example 4
What is the slope of a line through and ?
Use the formula to identify the slope of this line.
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.
C.
B.
A.
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, and .
The slope of the line in the diagram is 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 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 in the numerator and the given value in the denominator. The reciprocal of is and the reciprocal of is . 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 is and the opposite reciprocal of is .
Example 6
Which of the following could represent the slope of a line perpendicular to the one shown below?
D.
C.
B.
A.
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, and .
The slope of the line in the diagram is . 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 is . 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 and
Begin by plotting these points on a coordinate grid. Remember that the first number in the ordered pair represents the value and the second number represents the 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 and
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 and has a slope of
Begin by plotting the given point on a coordinate grid.
If the slope is , you can interpret that as . The fractional expression makes it easier to identify the rise and the run. So, the rise is and the run is . 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 You can connect these two points, extend the segment, and add arrows to show the line that passes through with a slope of
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.
Review Questions
Solve each problem.
Which term best describes the slope of the line below?
positive
negative
zero
undefined
Which term best describes the slope of the following line?
positive
negative
zero
undefined
What is the slope of the following line?
What would the slope be of a line parallel to the one following?
Which term best describes the slope of the following line?
positive
negative
zero
undefined
What is the slope of the following line?
Plot a line through the point below with a slope of What is the equation of that line?
Plot a line through the point below with a slope of .
What would the slope be of a line perpendicular to the line below?
Plot a line that travels through the point below and has a slope of
Review Answers
c
b
d
Equations of Lines
Learning Objectives
identify and write equations in slope-intercept form.
identify equations of parallel lines.
identify equations of perpendicular lines.
identify and write equations in standard form.
Introduction
Every line that you can represent graphically on the coordinate plane can also be represented algebraically. That means that you can create an equation relating and that corresponds to any graph of a straight line. In this lesson, you’ll learn how to create an equation from a graph or points given, identify equations of parallel and perpendicular lines, and practice using both slope-intercept and standard form.
Slope-Intercept Equations
The first type of linear equation to study is the most straightforward. It is called slope-intercept form and involves both the slope of the line and its intercept. A intercept is the point at which the line crosses the vertical axis. So, it will be the value of when is equal to . The generic formula for an equation in slope-intercept form is as follows.
In this equation, and remain as variables, is the slope of the line, and is the intercept of the line. So, if you know that a line has a slope of and it crosses the axis at its equation in slope-intercept form would be .
This form is especially useful for identifying the equation of a line given its graph. You already know how to deduce the slope by finding two points and using the slope formula. You can identify the intercept by sight by finding where the line crosses the axis on the graph. The value of is the coordinate of this point.
Example 1
Write an equation in slope-intercept form that represents the following line.
First find the slope of the line. You already know how to do this using the slope formula. In this scenario, pick two points on the line to complete the formula. Use and .
The slope of the line is . This value will replace in the slope-intercept equation. Now you need to find the intercept. Identify on the graph where the line intersects the axis. It crosses the axes at so the intercept is This will replace in the slope-intercept equation, so now you have all the information you need to write the full equation. The equation for the line shown in the graph is .
Equations of Parallel Lines
You studied parallel lines and their graphical relationships in the last lesson. In this lesson, you will learn how to easily identify equations of parallel lines. It’s simple—look for equations that have the same slope. As long as the intercepts are not the same and the slopes are equal, the lines are parallel. (If the intercept and the slope are the same, then the two equations would be for the same line, and a line cannot be parallel to itself.)
Example 2
Millicent drew the line below.
Which of the following equations could represent a line parallel to the one Millicent drew?
D.
C.
B.
A.
All you really need to do to solve this problem is identify the slope of the line in Millicent’s graph. Identify two points on the graph, and find the slope using the slope formula. Use points and
The slope of Millicent’s line is All you have to do is identify which equation among the four choices has a slope of You can disregard all other information. The only equation that has a slope of is choice C, so it is the correct answer.
Equations of Perpendicular Lines
You also studied perpendicular lines and their graphical relationships in the last lesson. Remember that the slopes of perpendicular lines are opposite reciprocals. In this lesson, you will learn how to easily identify equations of perpendicular lines. Look for equations that have the slopes that are opposite reciprocals of each other. In this case it doesn’t matter what the intercept is; as long as the slopes are opposite reciprocals, the lines are perpendicular.
Example 3
Kieran drew the line in this graph.
Which of the following equations could represent a line perpendicular to the one Kieran drew?
D.
C.
B.
A.
All you really need to do to solve this problem is identify the slope of the line in Kieran’s g
raph and find its opposite reciprocal. To begin, identify two points on the graph, and find the slope using the slope formula. Use points and .
The slope of Millicent’s line is . Now find the opposite reciprocal of this value. The reciprocal of is , and the opposite of is . So, is the opposite reciprocal of . Now find the equation that has a slope of . The only equation that has a slope of is choice D, so it is the correct answer.
Equations in Standard Form
There are other ways to write the equation of a line besides the slope intercept form. One alternative is standard form. Standard form is represented by the equation below.
In this equation, both and cannot be . Also, if possible, and should be integers.
Example 4
Convert the equation into standard form.
The goal is to remove the fractions and have and on the same side of the equals sign. To start, multiply the entire equation by 7 to eliminate the denominator of .
Next multiply the equation by to eliminate the denominator of .
Now add to both sides of the equation to get and on the same side.