Slopes of Parallel Lines Questions

A detailed tutorial on how to prove that the slopes of two parallel lines are equal. Questions on parallel lines are also included with their detailed solutions.

Prove that two parallel lines have equal slopes

• The figure below shows two parallel lines L1 and L2. Points A and B are on the line L1 and points C and D are on the line L2.
  

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• Let BM and DN be parallel to the y axis and AM and CN parallel to the x axis. Triangles ABM and CDN have all their corresponding sides parallel are therefore similar. Hence
  BM / DN = MA / NC
  
• The above may be written
  BM / MA = DN / NC
  
• According to the definition of slope, BM / MA is the slope of line L1 and DN / NC is the slope of L2 and are therefore equal.

Questions on parallel lines

Question 1
Which of the lines given by the equations
a) y = 2 x - 3    b) 2 x - y = 2    c) - 4 y + 2 x = 0    d) - 4 y + 8 x = 9
are parallel?
Solution to Question 1
We first need to write each line in slope intercept form y = m x + b and find its slope m.
a) y = 2 x - 3, slope = 2
b) 2 x - y = 2, solve for y, y = 2 x - 2, slope = 2
c) - 4 y + 2 x = 0, solve for y, y = (1 / 2) x, slope = 1 / 2
d) - 4 y + 8 x = 9, solve for y, y = 2 x - 9 / 4, slope = 2
The lines with equal slopes are the lines given in parts a) b) and d) and they are therefore parallel.

Question 2
Find k so that the lines with equations 6 k x - 3 y = 9 and - 4 x + 5 y = 7 are parallel.
Solution to Question 2
Solve for y and find the slope of each line
6 k x - 3 y = 9 , solve for y, y = 2 k x - 3 , slope = 2 k
- 4 x + 5 y = 7 , solve for y , y = (4 / 5) x + 7 / 5 , slope = 4 / 5
The two lines are parallel and therefore their slopes are equal; hence
2 k = 4 / 5
Solve for k
k = 2 / 5

Question 3
Find a so that the line through the points (1 , 2) and (0 , 3) and the line through the points (a , 2) and ( -2 , 7) are parallel.
Solution to Question 3
Find the slope m₁ through the points (1 , 2) and (0 , 3)
m₁ = (3 - 2) /(0 - 1) = - 1
Find the slope m₂ through the points (a , 2) and (-2 , 7)
m₂ = (7 - 2) / (-2 - a) = 5 / (-2 - a)
The two lines are parallel, hence their slopes are equal
5 / (-2 - a) = - 1
Solve for a
a = 3

Question 4
Find the coordinate of point D so that the points A(0 , 2) , B(2,6), C(8 , 8) and D are the vertices of a parallelogram.
Solution to Question 4
Let (a , b) be the coordinates of point D. For the points A, B, C and D to be the vertices of a parallelogram, segment AB must be parallel to segment DC and segment BC must be parallel to segment AD as shown in the figure below.

Question 5
Find the equation of a line L₁ that passes through the point (2 , 1) and is parallel to the line L₂ through the points (-1 , 2) and (3 , 4).
Solution to Question 5
The slope m₂ of the line L₂ through the points (-1 , 2) and (3 , 4) is given by
m₂ = (4 - 2) / (3 - (-1) ) = 2 / 4 = 1 / 2
Let m₁ be the slope of the line L₁. L₁ and L₂ being parallel their slopes are equal. Hence
m₁ = m₂ = 1 / 2
The equation of line L₁ in point (2 , 1) slope m₁ form is given by
y - 1 = (1 / 2)(x - 2)
which simplifies to
y = (1 / 2) x

Question 6
Find the equation of a line L₁ that passes through the point (-1 , 1) and is parallel to the line L₂ with equation 2 y + 4 x = 2.
Solution to Question 6
The slope m₂ of the line L₂ is found by first writing its equation in slope intercept form y = m x + b. Hence solve 2 y + 4 x = 2 for y to obtain
y = - 2 x + 1
The slope of L₂ is
m₂ = - 2
Since L₁ and L₂ are parallel their slopes are equal. Hence
m₁ = m₂ = - 2
The equation of line L₁ in point (-1 , 1) slope m₁ form is given by
y - 1 = - 2(x - (-1))
which simplifies to
y = - 2 x - 1

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