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Parallel and Perpendicular Slopes
8th - 10th , parallel and perpendicular lines, 3rd - 5th , 9th - 10th , lines, line segments, parallel, perpendi..., equations of parallel and perpendicular ..., 10th - 12th , slopes of parallel and perpendicular lin..., 7th - 9th .
2-4 Slopes of Parallel and Perpendicular...
8th - 11th grade, mathematics.
2-4 Slopes of Parallel and Perpendicular Lines
11 questions
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- 1. Multiple Choice Edit 5 minutes 1 pt Slopes of parallel lines are... opposite reciprocals. opposites. the same. always 7.
- 2. Multiple Choice Edit 5 minutes 1 pt Slopes of perpendicular lines are... opposite reciprocals. opposites. the same. always 7.
- 3. Multiple Choice Edit 5 minutes 1 pt The following represents slopes of two lines. Which answer choice describes the two lines? m = 5 and m = -1/5 Parallel Perpendicular Neither can not be determined
- 4. Multiple Choice Edit 5 minutes 1 pt The following represents slopes of two lines. Which answer choice describes the two lines? m = -1/3 and m = -1/3 Parallel Perpendicular Neither can not be determined
Which point on the x -axis lies on the line that passes through point C and is parallel to line AB?
Which point on the x -axis lies on the line that passes through point P and is perpendicular to line MN?
- 8. Multiple Choice Edit 5 minutes 1 pt A line contains the points (4, 3) and (19, −2). Which equation represents this line? y = -x + 7 y = -3x + 12 y = - 1/3x + 13/3 y = 1/3x + 5/3
- 9. Multiple Choice Edit 5 minutes 1 pt What is the equation of a line that is perpendicular to this line and goes through the given point? y = 3x + 2 (3, -4) y = 3x - 2 y = -1/3x - 5 y = 3x - 4 y = -1/3x - 3
Which point is on the line that passes through point H and is perpendicular to line FG?
Write the equation of the line PARALLEL to the line y = 4x - 9 that passes through the point (-2, 4)
y = 4x + 12
y = -1/4x + 5/2
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Parallel & Perpendicular Lines
Slope & Formula Hor. & Vert. Lines Par. & Perp. Lines
Parallel lines and their slopes are easy. Since slope is a measure of the angle of a line from the horizontal, and since parallel lines must have the same angle, then parallel lines have the same slope — and lines with the same slope are parallel.
Perpendicular lines are a bit more complicated.
If you visualize a line with positive slope (so it's an increasing line), then the perpendicular line must have negative slope (because it will have to be a decreasing line). So perpendicular lines have slopes which have opposite signs.
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Parallel and Perpendicular Lines
The other "opposite" thing with perpendicular slopes is that their values are reciprocals; that is, you take the one slope value, and flip it upside down. (This is the non -obvious thing about the slopes of perpendicular lines.) Put this together with the sign change, and you get that the slope of a perpendicular line is the "negative reciprocal" of the slope of the original line — and two lines with slopes that are negative reciprocals of each other are perpendicular to each other.
In your homework, you will probably be given some pairs of points, and be asked to state whether the lines through the pairs of points are "parallel, perpendicular, or neither". To answer the question, you'll have to calculate the slopes and compare them. Here's how that works:
One line passes through the points (−1, −2) and (1, 2) ; another line passes through the points (−2, 0) and (0, 4) . Are these lines parallel, perpendicular, or neither?
To answer this question, I'll find the two slopes. So I can keep things straight and tell the difference between the two slopes, I'll use subscripts.
Since these two lines have identical slopes, then:
these lines are parallel.
One line passes through the points (0, −4) and (−1, −7) ; another line passes through the points (3, 0) and (−3, 2) . Are these lines parallel, perpendicular, or neither?
I'll find the values of the slopes.
the lines are perpendicular.
One line passes through the points (−4, 2) and (0, 3) ; another line passes through the points (−3, −2) and (3, 2) . Are these lines parallel, perpendicular, or neither?
I'll find the slopes.
These slope values are not the same, so the lines are not parallel. The slope values are also not negative reciprocals, so the lines are not perpendicular. Then the answer is:
these lines are neither.
Find the slope of a line perpendicular to the line y = −4 x + 9 .
They've given me the original line's equation, and it's in " y = " form, so it's easy to find the slope. I can just read the value off the equation: m = −4 .
This slope can be turned into a fraction by putting it over 1 , so this slope can be restated as:
m = −4 / 1
To get the negative reciprocal, I need to flip this fraction, and change the sign. Then the slope of any line perpendicular to the given line is:
Warning: When asked a question of this type ("are these lines parallel or perpendicular?"), do not start drawing pictures. If the lines are close to being parallel or close to being perpendicular (or if you draw the lines messily), you can very-easily get the wrong answer from your picture.
Besides, they're not asking if the lines look parallel or perpendicular; they're asking if the lines actually are parallel or perpendicular. The only way to be sure of your answer is to do the algebra.
URL: https://www.purplemath.com/modules/slope3.htm
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Parallel and Perpendicular Lines
How to use Algebra to find parallel and perpendicular lines
Parallel Lines
How do we know when two lines are parallel ?
Their slopes are the same!
Find the equation of the line that is:
- parallel to y = 2x + 1
- and passes though the point (5,4)
The slope of y = 2x + 1 is 2
The parallel line needs to have the same slope of 2.
We can solve it by using the "point-slope" equation of a line :
y − y 1 = 2(x − x 1 )
And then put in the point (5,4):
y − 4 = 2(x − 5)
That is an answer!
But it might look better in y = mx + b form. Let's expand 2(x − 5) and then rearrange:
y − 4 = 2x − 10
Vertical Lines
But this does not work for vertical lines ... I explain why at the end.
Not The Same Line
Be careful! They may be the same line (but with a different equation), and so are not parallel .
How do we know if they are really the same line? Check their y-intercepts (where they cross the y-axis) as well as their slope:
Example: is y = 3x + 2 parallel to y − 2 = 3x ?
For y = 3x + 2 : the slope is 3, and y-intercept is 2
For y − 2 = 3x : the slope is 3, and y-intercept is 2
So they are the same line and so are not parallel
Perpendicular Lines
Two lines are perpendicular when they meet at a right angle (90°).
To find a perpendicular slope:
When one line has a slope of m , a perpendicular line has a slope of −1 m
In other words the negative reciprocal
Find the equation of the line that is
- perpendicular to y = −4x + 10
- and passes though the point (7,2)
The slope of y = −4x + 10 is −4
The negative reciprocal of that slope is:
m = −1 −4 = 1 4
So the perpendicular line will have a slope of 1/4:
y − y 1 = (1/4)(x − x 1 )
And now we put in the point (7,2):
y − 2 = (1/4)(x − 7)
That answer is OK, but let's also put it in "y=mx+b" form:
y − 2 = x/4 − 7/4
y = x/4 + 1/4
Quick Check of Perpendicular
When we multiply a slope m by its perpendicular slope −1 m we get simply −1 .
So to quickly check if two lines are perpendicular:
When we multiply their slopes, we get −1
Are these two lines perpendicular?
When we multiply the two slopes we get:
2 × (−0.5) = −1
Yes, we got −1, so they are perpendicular.
The previous methods work nicely except for a vertical line :
In this case the gradient is undefined (as we cannot divide by 0 ):
m = y A − y B x A − x B = 4 − 1 2 − 2 = 3 0 = undefined
So just rely on the fact that:
- a vertical line is parallel to another vertical line.
- a vertical line is perpendicular to a horizontal line (and vice versa).
- parallel lines: same slope
- perpendicular lines: negative reciprocal slope (−1/m)
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We have an expert-written solution to this problem! Study with Quizlet and memorize flashcards containing terms like parallel lines, perpendicular lines, Theorem 2-13 Two lines are parallel if and only if ... and more.
Study with Quizlet and memorize flashcards containing terms like What is the slope of line PQ? What is the slope of line MN? How are the two lines related?, Which line is perpendicular to a line that has a slope of -1/3?, Which points could be on the line that is parallel to and passes through point J? Check all that apply. and more.
Figure 8.11.1. Solution. Looking at ℓ1, we can start at (− 3, 1) and reach the next point at (0, − 1). We see that we will move down two units and run to the right 3 units. Hence, ℓ1 ’s slope is − 2 3. Now let’s look at ℓ2 and obtain its slope. We will start at (0, 2) and reach the next point at (3, 0).
y = mx + b. slope formula. m=y2-y1/x2-x1. Study with Quizlet and memorize flashcards containing terms like Parallel Non-Vertical Lines Theorem, Vertical Lines Theorem, Perpendicular Non-Vertical Lines Theorem and more.
Learn how to tell if two distinct lines are parallel, perpendicular, or neither. Use the slope formula to calculate the slope of each line to determine if they are parallel, perpendicular, or neither.
Definition of Parallel and Perpendicular. Parallel lines are lines in the same plane that never intersect. Two nonvertical lines in the same plane, with slopes \ (m_ {1}\) and \ (m_ {2}\), are parallel if their slopes are the same, \ (m_ {1}=m_ {2}\). Consider the following two lines:
2-4 Slopes of Parallel and Perpendicular Lines quiz for 8th grade students. Find other quizzes for Mathematics and more on Quizizz for free!
Demonstrates how to determine if slopes are for parallel lines, perpendicular lines, or neither. Explains why graphing is not generally helpful for this type of question.
Parallel and Perpendicular Lines. PearsonRealize.com. Write an equation for the line that passes through the given point and is parallel to the graph of the given equation. 1. y = 3x − 2; (3, 2) y = 3x − 7. 2. y = __ 2 . 3 x + 19; (−9, 4) y = __ 2 . 3 x + 10. 3. 3x + 4y = 12; (−4, 7) __ 3 y = − x + 4. 4.
a vertical line is parallel to another vertical line. a vertical line is perpendicular to a horizontal line (and vice versa). Summary. parallel lines: same slope; perpendicular lines: negative reciprocal slope (−1/m)