Worksheet: Equations of Parallel and Perpendicular Lines

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.

  • A𝑦=βˆ’6π‘₯βˆ’7
  • B𝑦=βˆ’6π‘₯βˆ’5
  • C𝑦=6π‘₯+5
  • D𝑦=βˆ’16π‘₯+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).

  • A𝑦=38π‘₯+258
  • B𝑦=38π‘₯βˆ’258
  • C𝑦=βˆ’6π‘₯+83
  • D𝑦=βˆ’83π‘₯+6
  • E𝑦=βˆ’83π‘₯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).

  • A𝑦=12π‘₯+52
  • B𝑦=βˆ’2π‘₯+5
  • C𝑦=βˆ’2π‘₯+7
  • D𝑦=2π‘₯βˆ’1
  • E𝑦=βˆ’12π‘₯+72

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.

  • A𝑦=π‘₯+1
  • B𝑦=π‘₯βˆ’53
  • C𝑦=βˆ’π‘₯+3
  • D𝑦=π‘₯+13

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.

  • A𝑦=βˆ’3π‘₯βˆ’3
  • B𝑦=βˆ’3π‘₯βˆ’9
  • C𝑦=3π‘₯+9
  • D𝑦=βˆ’13π‘₯+3

Q6:

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

  • A𝑦=βˆ’109π‘₯+559
  • B𝑦=βˆ’109π‘₯βˆ’559
  • C𝑦=4110π‘₯βˆ’910
  • D𝑦=910π‘₯βˆ’4110
  • E𝑦=910π‘₯+4110

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).

  • A𝑦=8π‘₯+13
  • B𝑦=8π‘₯βˆ’13
  • C𝑦=398π‘₯+18
  • D𝑦=βˆ’18π‘₯βˆ’398
  • E𝑦=βˆ’18π‘₯+398

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).

  • A𝑦=βˆ’12π‘₯βˆ’32
  • B𝑦=2π‘₯βˆ’9
  • C𝑦=βˆ’12π‘₯+32
  • D𝑦=βˆ’32π‘₯+12
  • E𝑦=2π‘₯+9

Q9:

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

  • Aβˆ’2
  • B132
  • Cβˆ’6
  • D16

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).

  • A𝑦=βˆ’1916π‘₯+13516
  • B𝑦=1619π‘₯+34119
  • C𝑦=βˆ’34119π‘₯+1619
  • D𝑦=βˆ’1916π‘₯βˆ’13516
  • E𝑦=1619π‘₯βˆ’34119

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.

  • A𝑦=π‘₯βˆ’4
  • B𝑦=βˆ’74π‘₯+1
  • C𝑦=βˆ’4π‘₯+1
  • D𝑦=47π‘₯
  • E𝑦=47π‘₯+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.

  • A3π‘₯+10𝑦+47=0
  • B3π‘₯βˆ’10π‘¦βˆ’47=0
  • C10π‘₯+3𝑦+25=0
  • D13π‘₯+10𝑦+63=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.

  • A𝑦=56π‘₯+15
  • B𝑦=βˆ’65π‘₯+4
  • C𝑦=3π‘₯βˆ’1
  • D𝑦=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?

  • A𝑦=βˆ’1
  • Bπ‘₯=βˆ’1
  • C𝑦=0
  • Dπ‘₯=4
  • E𝑦=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.

  • Aβˆ’23
  • B13
  • Cβˆ’34
  • D43

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?

  • A3𝑦=1βˆ’19π‘₯
  • B3π‘₯βˆ’19𝑦=5
  • C2βˆ’19𝑦=3π‘₯
  • D3+19𝑦=2π‘₯
  • E3𝑦=19π‘₯+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.

  • A𝑦=34π‘₯βˆ’14
  • B𝑦=βˆ’14π‘₯+32
  • C𝑦=βˆ’14π‘₯+34
  • D𝑦=4π‘₯βˆ’12

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).

  • A𝑦=βˆ’513π‘₯βˆ’7913
  • B𝑦=135π‘₯βˆ’8
  • C𝑦=βˆ’513π‘₯βˆ’21
  • D𝑦=135π‘₯βˆ’21
  • E𝑦=βˆ’135π‘₯βˆ’7913

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).

  • A𝑦=54π‘₯+94
  • B𝑦=βˆ’2π‘₯βˆ’1
  • C𝑦=βˆ’2π‘₯+3
  • D𝑦=βˆ’45π‘₯βˆ’15

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?

  • A𝑦=32(π‘₯+1)βˆ’4
  • B𝑦=32(π‘₯βˆ’1)βˆ’4
  • C𝑦=βˆ’32(π‘₯+1)+4
  • D𝑦=βˆ’(π‘₯βˆ’1)+4
  • E𝑦=βˆ’32(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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