Slopes of Parallel Lines Questions

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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