Worksheet: Equations of Parallel and Perpendicular Lines

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In this worksheet, we will practice writing the equation of a line parallel or perpendicular to another line.

Q1:

Write, in the form 𝑦=π‘šπ‘₯+𝑐, the equation of the line through (βˆ’1,βˆ’1) that is parallel to the line βˆ’6π‘₯βˆ’π‘¦+4=0.

  • A 𝑦 = βˆ’ 6 π‘₯ βˆ’ 7
  • B 𝑦 = βˆ’ 6 π‘₯ βˆ’ 5
  • C 𝑦 = 6 π‘₯ + 5
  • D 𝑦 = βˆ’ 1 6 π‘₯ + 7

Q2:

Find, in slope-intercept form, the equation of the line parallel to 𝑦=βˆ’83π‘₯+3 that passes through point 𝐴(βˆ’3,2).

  • A 𝑦 = 3 8 π‘₯ + 2 5 8
  • B 𝑦 = 3 8 π‘₯ βˆ’ 2 5 8
  • C 𝑦 = βˆ’ 6 π‘₯ + 8 3
  • D 𝑦 = βˆ’ 8 3 π‘₯ + 6
  • E 𝑦 = βˆ’ 8 3 π‘₯ βˆ’ 6

Q3:

A straight line 𝐿 has the equation 𝑦=βˆ’2π‘₯βˆ’3. Find the equation of the line parallel to 𝐿 that passes through the point (1,3).

  • A 𝑦 = 1 2 π‘₯ + 5 2
  • B 𝑦 = βˆ’ 2 π‘₯ + 5
  • C 𝑦 = βˆ’ 2 π‘₯ + 7
  • D 𝑦 = 2 π‘₯ βˆ’ 1
  • E 𝑦 = βˆ’ 1 2 π‘₯ + 7 2

Q4:

Write, in the form 𝑦=π‘šπ‘₯+𝑐, the equation of the line through (1,2) that is parallel to the line 3π‘₯βˆ’3𝑦+7=0.

  • A 𝑦 = π‘₯ + 1
  • B 𝑦 = π‘₯ βˆ’ 5 3
  • C 𝑦 = βˆ’ π‘₯ + 3
  • D 𝑦 = π‘₯ + 1 3

Q5:

Write, in the form 𝑦=π‘šπ‘₯+𝑐, the equation of the line through (βˆ’2,3) that is parallel to the line βˆ’3π‘₯βˆ’π‘¦+9=0.

  • A 𝑦 = βˆ’ 3 π‘₯ βˆ’ 3
  • B 𝑦 = βˆ’ 3 π‘₯ βˆ’ 9
  • C 𝑦 = 3 π‘₯ + 9
  • D 𝑦 = βˆ’ 1 3 π‘₯ + 3

Q6:

Find, in slope-intercept form, the equation of the line parallel to 𝑦=910π‘₯+4 that passes through point 𝐴(1,5).

  • A 𝑦 = βˆ’ 1 0 9 π‘₯ + 5 5 9
  • B 𝑦 = βˆ’ 1 0 9 π‘₯ βˆ’ 5 5 9
  • C 𝑦 = 4 1 1 0 π‘₯ βˆ’ 9 1 0
  • D 𝑦 = 9 1 0 π‘₯ βˆ’ 4 1 1 0
  • E 𝑦 = 9 1 0 π‘₯ + 4 1 1 0

Q7:

Find, in slope-intercept form, the equation of the line parallel to 𝑦=βˆ’18π‘₯+4 that passes through point 𝐴(βˆ’1,5).

  • A 𝑦 = 8 π‘₯ + 1 3
  • B 𝑦 = 8 π‘₯ βˆ’ 1 3
  • C 𝑦 = 3 9 8 π‘₯ + 1 8
  • D 𝑦 = βˆ’ 1 8 π‘₯ βˆ’ 3 9 8
  • E 𝑦 = βˆ’ 1 8 π‘₯ + 3 9 8

Q8:

Find, in slope-intercept form, the equation of the line perpendicular to 𝑦=2π‘₯βˆ’4 that passes through the point 𝐴(3,βˆ’3).

  • A 𝑦 = βˆ’ 1 2 π‘₯ βˆ’ 3 2
  • B 𝑦 = 2 π‘₯ βˆ’ 9
  • C 𝑦 = βˆ’ 1 2 π‘₯ + 3 2
  • D 𝑦 = βˆ’ 3 2 π‘₯ + 1 2
  • E 𝑦 = 2 π‘₯ + 9

Q9:

Suppose that the points 𝐴(βˆ’3,βˆ’1), 𝐡(1,2), and 𝐢(7,𝑦) form a right-angled triangle at 𝐡. What is the value of 𝑦?

  • A βˆ’ 2
  • B 1 6
  • C βˆ’ 6
  • D 1 3 2

Q10:

Given that the coordinates of the points 𝐴, 𝐡, 𝐢, and 𝐷 are (βˆ’15,8), (βˆ’6,10), (βˆ’8,βˆ’7), and (βˆ’6,βˆ’16), respectively, determine whether ⃖⃗𝐴𝐡 and ⃖⃗𝐢𝐷 are parallel, perpendicular, or neither.

  • Aperpendicular
  • Bneither
  • Cparallel

Q11:

Determine, in slope-intercept form, the equation of the line passing through 𝐴(13,βˆ’7) perpendicular to the line passing through 𝐡(8,βˆ’9) and 𝐢(βˆ’8,10).

  • A 𝑦 = βˆ’ 1 9 1 6 π‘₯ + 1 3 5 1 6
  • B 𝑦 = 1 6 1 9 π‘₯ + 3 4 1 1 9
  • C 𝑦 = βˆ’ 3 4 1 1 9 π‘₯ + 1 6 1 9
  • D 𝑦 = βˆ’ 1 9 1 6 π‘₯ βˆ’ 1 3 5 1 6
  • E 𝑦 = 1 6 1 9 π‘₯ βˆ’ 3 4 1 1 9

Q12:

Write, in the form 𝑦=π‘šπ‘₯+𝑐, the equation of the line that is parallel to the line βˆ’4π‘₯+7π‘¦βˆ’4=0 and that intercepts the 𝑦-axis at 1.

  • A 𝑦 = π‘₯ βˆ’ 4
  • B 𝑦 = βˆ’ 7 4 π‘₯ + 1
  • C 𝑦 = βˆ’ 4 π‘₯ + 1
  • D 𝑦 = 4 7 π‘₯
  • E 𝑦 = 4 7 π‘₯ + 1

Q13:

If 𝐴(3,βˆ’1) and 𝐡(βˆ’4,βˆ’8), find the cartesian equation of the straight line passing through the point of division of 𝐴𝐡 internally in the ratio 4∢3 and perpendicular to the straight line whose equation is 10π‘₯+3π‘¦βˆ’65=0.

  • A 3 π‘₯ + 1 0 𝑦 + 4 7 = 0
  • B 3 π‘₯ βˆ’ 1 0 𝑦 βˆ’ 4 7 = 0
  • C 1 0 π‘₯ + 3 𝑦 + 2 5 = 0
  • D 1 3 π‘₯ + 1 0 𝑦 + 6 3 = 0

Q14:

Consider the triangle on 𝐴(βˆ’6,9), 𝐡(4,βˆ’3), and 𝐢(1,βˆ’6), and let 𝐷 be the midpoint of 𝐴𝐡. Now let 𝐸 on 𝐴𝐢 be the intersection of the parallel to ⃖⃗𝐡𝐢 through the point 𝐷. Find the equation of ⃖⃗𝐷𝐸 in the form 𝑦=π‘šπ‘₯+𝑐.

  • A 𝑦 = 5 6 π‘₯ + 1 5
  • B 𝑦 = βˆ’ 6 5 π‘₯ + 4
  • C 𝑦 = 3 π‘₯ βˆ’ 1
  • D 𝑦 = π‘₯ + 4

Q15:

Lines 𝐴 and 𝐡 are perpendicular to each other and meet at (βˆ’1,4). If the slope of 𝐴 is 0, what is the equation of line 𝐡?

  • A 𝑦 = βˆ’ 1
  • B π‘₯ = βˆ’ 1
  • C 𝑦 = 0
  • D π‘₯ = 4
  • E 𝑦 = 4

Q16:

Determine whether the lines 𝑦=βˆ’17π‘₯βˆ’5 and 𝑦=βˆ’17π‘₯βˆ’1 are parallel, perpendicular, or neither.

  • Aperpendicular
  • Bparallel
  • Cneither

Q17:

If a line 𝐿 is perpendicular to the line βˆ’2𝑦+10=βˆ’6π‘₯+7, and 𝐿 passes through the points 𝐴(𝑛,βˆ’10) and 𝐡(βˆ’7,2), what is the value of 𝑛?

Q18:

Suppose that 𝐿 is the line π‘Žπ‘₯βˆ’π‘¦+15=0, and 𝐿 the line βˆ’2π‘₯3+𝑦2=βˆ’23. Find the value of π‘Ž so that 𝐿βˆ₯𝐿.

  • A βˆ’ 2 3
  • B 1 3
  • C βˆ’ 3 4
  • D 4 3

Q19:

If the two straight lines πΏβˆΆβˆ’8π‘₯+7π‘¦βˆ’9=0 and πΏβˆΆπ‘Žπ‘₯+24𝑦+56=0 are perpendicular, find the value of π‘Ž.

Q20:

Which of the following lines is perpendicular to the line 19π‘₯βˆ’3𝑦=5?

  • A 3 𝑦 = 1 βˆ’ 1 9 π‘₯
  • B 3 π‘₯ βˆ’ 1 9 𝑦 = 5
  • C 2 βˆ’ 1 9 𝑦 = 3 π‘₯
  • D 3 + 1 9 𝑦 = 2 π‘₯
  • E 3 𝑦 = 1 9 π‘₯ + 4

Q21:

Given 𝐴(4,4) and 𝐡(2,βˆ’4), find the equation of the perpendicular to 𝐴𝐡 that passes through the midpoint of this line segment. Give your answer in the form 𝑦=π‘šπ‘₯+𝑐.

  • A 𝑦 = 3 4 π‘₯ βˆ’ 1 4
  • B 𝑦 = βˆ’ 1 4 π‘₯ + 3 2
  • C 𝑦 = βˆ’ 1 4 π‘₯ + 3 4
  • D 𝑦 = 4 π‘₯ βˆ’ 1 2

Q22:

Write, in the form 𝑦=π‘šπ‘₯+𝑐, the equation of the line through 𝐴(5,βˆ’8) that is perpendicular to 𝐴𝐡, where 𝐡(βˆ’8,βˆ’3).

  • A 𝑦 = βˆ’ 5 1 3 π‘₯ βˆ’ 7 9 1 3
  • B 𝑦 = 1 3 5 π‘₯ βˆ’ 8
  • C 𝑦 = βˆ’ 5 1 3 π‘₯ βˆ’ 2 1
  • D 𝑦 = 1 3 5 π‘₯ βˆ’ 2 1
  • E 𝑦 = βˆ’ 1 3 5 π‘₯ βˆ’ 7 9 1 3

Q23:

Find the equation of the straight line passing through the point (βˆ’1,1) and perpendicular to the straight line passing through the points (βˆ’9,9) and (6,βˆ’3).

  • A 𝑦 = 5 4 π‘₯ + 9 4
  • B 𝑦 = βˆ’ 2 π‘₯ βˆ’ 1
  • C 𝑦 = βˆ’ 2 π‘₯ + 3
  • D 𝑦 = βˆ’ 4 5 π‘₯ βˆ’ 1 5

Q24:

Lines 𝐴 and 𝐡 are perpendicular to each other and meet at (1,4). If the slope of 𝐴 is 32, what is the equation of line 𝐡?

  • A 𝑦 = 3 2 ( π‘₯ + 1 ) βˆ’ 4
  • B 𝑦 = 3 2 ( π‘₯ βˆ’ 1 ) βˆ’ 4
  • C 𝑦 = βˆ’ 3 2 ( π‘₯ + 1 ) + 4
  • D 𝑦 = βˆ’ ( π‘₯ βˆ’ 1 ) + 4
  • E 𝑦 = βˆ’ 3 2 ( π‘₯ βˆ’ 1 ) + 4

Q25:

The straight lines 8π‘₯+5𝑦=8 and 8π‘₯+π‘Žπ‘¦=βˆ’8 are parallel. What is the value of π‘Ž?

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