6.4 Parallel and Perpendicular Lines

Parallel lines are lines that never intersect regardless of how far they are extended in either direction. Perpendicular lines are two lines that intersect at a right angle (90°). In this lesson, we will find the relationship between the slopes of two parallel lines and the relationship between the slopes of two perpendicular lines. Answer the following questions to explore these relationships.

A coordinate plane. Both axes have a minimum of -4 and maximum of 4. and a scale of 1. Two lines are graphed: y=2/3x+2 and y=2/3x-1 a) What kind of lines are these?

 

b) Find the slope of the top line.

 

c) What do you think the slope of the bottom line will be?

 

d) Find the slope of the bottom line.

 

e) What can we say about the slope of parallel lines?

 
A coordinate plane. x axis has a minimum of -4 and maximum of 2. and a scale of 1. y axis has a minimum of -4 and maximum of 1. and a scale of 1. Two lines are graphed: y=-4x+1 and y=1/4x-2 and then intersect at a 90 degree angle a) What kind of lines are these?

 

b) Find the slope of both lines.

 

c) What do you notice about these two slopes?

 
Table 6.4.1
Slope of Parallel Lines
The slopes of two parallel lines, m_1 and m_2, are equal to each other.

m_1 = m_2
Table 6.4.2
Slope of Perpendicular Lines
The slopes of two perpendicular lines, m_1 and m_2, are negative reciprocals of each other and have a product of –1.

m_1 = -\dfrac{1}{m_2}

{m_1}\times{m_2} = -1
Practice 1

Consider the slopes below.
i) Write the slope of a line that would be parallel to the given line.
ii) Write the slope of a line that would be perpendicular to the given line.

a) m=\dfrac{2}{5}

 

 

 b) m=-\dfrac{4}{3}

 

 

 c) m=-5

 

 
Practice 2

Consider the graphs below.
i) Write the slope of a line that would be parallel to the given line.
ii) Write the slope of a line that would be perpendicular to the given line.

a)
The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The line goes through the points (0, negative 5) and (1, negative 2).		 b)
The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The line goes through the points (0, 1) and (3, negative 3).
Practice 3

Consider the line of each set of ordered pairs .
i) Write the slope of a line that would be parallel to the given line.
ii) Write the slope of a line that would be perpendicular to the given line.

a) (–1, 5) and (–7, 4)

 

 

 b) (4, –2) and (2, 6)

 

 
Practice 4

Line AB goes through the points A (3, -6) and B (7, n). Line AB is perpendicular to line CD. If line CD has a slope of -\dfrac{4}{3}, what is the the value of n?

 

 
Practice 5

Given that two parallel lines have the slopes -\dfrac{1}{2} and \dfrac{n}{10}, what is the value of n?

 

 
Practice 6

Given that two perpendicular lines have the slopes \dfrac{3}{5} and \dfrac{4}{n}, what is the value of n?

 

 

Homework 

  1. Consider the slopes below.
    i) Write the slope of a line that would be parallel to the given line.
    ii) Write the slope of a line that would be perpendicular to the given line.

    a) m=\dfrac{1}{3}

     

     

    		 b) m=-\dfrac{6}{5}

     

     

    		 c) m=-4

     

     
    d) m=\dfrac{1}{9}

     

     

    		 e) m=-\dfrac{7}{2}

     

     

    		 f) m=2

     

     
  2. Consider the graphs below.
    i) Write the slope of a line that would be parallel to the given line.
    ii) Write the slope of a line that would be perpendicular to the given line.

    a)
    The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The line goes through the points (0, negative 2) and (1, 2).    		 b)
    The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The line goes through the points (0,1) and (1, negative 2).    		 c)
    The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The line goes through the points (0, 4) and (1, 3).
    d)
    The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The line goes through the points (0, 3) and (1, 5).    		 e)
    The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The line goes through the points (-4,0) and (0,2).    		 f)
    The figure shows a line graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 12 to 12. The line goes through the points (-12,8) and (1,-12)).
  3. Consider the line of each set of ordered pairs .
    i) Write the slope of a line that would be parallel to the given line.
    ii) Write the slope of a line that would be perpendicular to the given line.

    a) (9, 7) and (7, 4)

     

     

     

    		 b) (8, –1) and (–1, –5)

     

     

     
    c) (–6, –7) and (–3, –4)

     

     

     

    		 d) (6, –2) and (3, 5)

     

     

     
    e) (1, 0) and (5, 6)

     

     

     

    		 f) (12, 26) and (–55, –62)

     

     

     
  4. Consider the following relations.
    i) Write the slope of a line that would be parallel to the given relation.
    ii) Write the slope of a line that would be perpendicular to the given relation.

    a)
    Horizontal line passing through y=-2.    		 b)
    Vertical line passing through x=3.
    c) (–2, 0) and (5, 0)

     

     

     

    		 d) (0, –1) and (0, –5)

     

     

     

  5. Find the unknown value in each question using the information provided.

    a) Line CD and line EF are parallel. Line CD goes through the points (1, –4) and (6, n), and mEF =\dfrac{2}{5}.

     

     
    b) Line PQ and line RS are perpendicular. Line PQ goes through the points (2, n) and (7, 1), and mRS = \dfrac{4}{3}.

     

     
    c) Line AB and line XY are parallel. Line AB passes through the points (n, –2) and (2, 3), and mXY = \dfrac{4}{5}.

     

     
    d) Line KL and line MN are perpendicular. Line KL passes through the points (0, –1) and (n, 4), and mMN = \dfrac{3}{2}

     

     

  6. Indicate whether the following statements are True or False.

    a) A line segment goes through points (1, 2) and (4, 6), and another line segment goes through points (2, –1) and (5, 3).

    True or False: These lines are parallel.

     

     
    b) A line segment goes through points (1, 2) and (3, 6), and another line segment goes through points (0, –1) and (1, 0).

    True or False: These lines are perpendicular.

     

     
    c) A line segment goes through points (0, 3) and (2, 5), and another line segment goes through points (4, –4) and (6, 0).

    True or False: These lines are parallel.

     

     
    d) A line segment goes through points (1, 2) and (3, 4), and another line segment goes through points (0, 0) and (1, –1).

    True or False: These lines are perpendicular.

     

     

  7. Find the unknown value in each question using the information provided.

    a) Given that two parallel lines have the slopes \dfrac{n}{4} and -\dfrac{1}{2}, what is the value of n?

     

     

     
    b) Given that two parallel lines have the slopes \dfrac{3}{5} and \dfrac{9}{n}, what is the value of n?

     

     

     
    c) Given that two parallel lines have the slopes \dfrac{10}{n} and \dfrac{4}{3}, what is the value of n?

     

     

     
    d) Given that two parallel lines have the slopes -\dfrac{7}{12} and \dfrac{n}{18}, what is the value of n?

     

     

     

  8. Find the unknown value in each question using the information provided.

    a) Given that two perpendicular lines have the slopes \dfrac{3}{5} and \dfrac{4}{n}, what is the value of n?

     

     

     
    b) Given that two perpendicular lines have the slopes \dfrac{n}{6} and -\dfrac{2}{3}, what is the value of n?

     

     

     
    c) Given that two perpendicular lines have the slopes \dfrac{5}{n} and \dfrac{3}{4}, what is the value of n?

     

     

     
    d) Given that two perpendicular lines have the slopes \dfrac{7}{n} and -\dfrac{5}{8}, what is the value of n?

     

     

     

Answers

1.

a) i) mparallel = \dfrac{1}{3}
ii) mperpendicular = –3		 b) i) mparallel = -\dfrac{6}{5}
ii) mperpendicular = \dfrac{5}{6}		 c) i) mparallel = –4
ii) mperpendicular = \dfrac{1}{4}
d) i) mparallel = \dfrac{1}{9}
ii) mperpendicular = –9		 e) i) mparallel = -\dfrac{7}{2}
ii) mperpendicular = \dfrac{2}{7}		 f) i) mparallel = 2
ii) mperpendicular = -\dfrac{1}{2}

2.

a) i) mparallel = 4
ii) mperpendicular = -\dfrac{1}{4}		 b) i) mparallel = –3
ii) mperpendicular = \dfrac{1}{3}		 c) i) mparallel = –1
ii) mperpendicular = 1
d) i) mparallel = -\dfrac{2}{5}
ii) mperpendicular = \dfrac{5}{2}		 e) i) mparallel = \dfrac{1}{2}
ii) mperpendicular = –2		 f) i) mparallel = -\dfrac{10}{11}
ii) mperpendicular = \dfrac{11}{10}

3.

a) i) mparallel = \dfrac{3}{2}
ii) mperpendicular = -\dfrac{2}{3}		 b) i) mparallel = \dfrac{4}{9}
ii) mperpendicular = -\dfrac{9}{4}		 c) i) mparallel = 1
ii) mperpendicular = –1
d) i) mparallel = -\dfrac{7}{3}
ii) mperpendicular = \dfrac{3}{7}		 e) i) mparallel = \dfrac{3}{2}
ii) mperpendicular = -\dfrac{2}{3}		 f) i) mparallel = \dfrac{88}{67}
ii) mperpendicular = -\dfrac{67}{88}

4.

a) i) mparallel = 0
ii) mperpendicular = undefined		 b) i) mparallel = undefined
ii) mperpendicular = 0
c) i) mparallel = 0
ii) mperpendicular = undefined		 d) i) mparallel = undefined
ii) mperpendicular = 0

5.

a) n = –2 b) n = \dfrac{19}{4} c) n = -\dfrac{17}{4} d) n = -\dfrac{15}{2}

6.

a) T b) F c) F d) T

7.

a) n = –2 b) n = 15 c) n = 7.5 d) n = –10.5

8.

a) n = -\dfrac{12}{5} b) n = 9 c) n = -\dfrac{15}{4} d) n = \dfrac{35}{8}

Attribution

Unless otherwise indicated, material on this page has been adapted from the following resource:

Mazur, I. (2021). Introductory algebra. BCcampus. https://collection.bccampus.ca/textbooks/intermediate-algebra-bccampus-412/, licensed under CC BY 4.0

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