Worksheet: Distance and Midpoints on the Complex Plane

In this worksheet, we will practice finding the distance and midpoint of two complex numbers in the complex plane.

Q1:

Find the midpoint of 3 + 5 𝑖 and 7 1 3 𝑖 .

  • A 5 4 𝑖
  • B 4 + 5 𝑖
  • C 2 + 9 𝑖
  • D 2 + 9 𝑖
  • E 1 0 8 𝑖

Q2:

A complex number 𝑤 lies at a distance of 5 2 from 𝑧 = 3 + 5 𝑖 and at a distance of 4 5 from 𝑧 = 6 2 𝑖 . Is the triangle formed by the points 𝑤 , 𝑧 , and 𝑧 a right triangle?

  • AYes
  • BNo

Q3:

What is the general form for the midpoint of two complex numbers 𝑧 and 𝑧 ?

  • A 2 ( 𝑧 𝑧 )
  • B 𝑧 + 𝑧 2
  • C 𝑧 + 𝑧 2
  • D 𝑧 𝑧 2
  • E 𝑧 𝑧 2

Q4:

What complex number lies at the midpoint of the complex numbers 𝑟 = 5 + 3 𝑖 and 𝑠 = 3 + 7 𝑖 when they are represented on a complex plane?

  • A 4 + 5 𝑖
  • B 2 + 4 𝑖
  • C 2 + 𝑖
  • D 1 + 2 𝑖
  • E 8 + 1 0 𝑖

Q5:

Find the complex number 𝑧 such that 4 + 3 𝑖 lies at the midpoint of 𝑧 and 3 4 𝑖 when they are represented on a complex plane.

  • A 1 1 + 2 𝑖
  • B 5 + 1 0 𝑖
  • C 5 + 1 3 𝑖
  • D 7 𝑖
  • E 1 + 7 𝑖

Q6:

Find the possible real values of 𝑏 such that the distance between the complex number 6 + 7 𝑖 and the complex number 3 + 𝑏 𝑖 is 5.

  • A 1.7 or 15.7
  • B 3 or 11
  • C 0.5 or 14.5
  • D 5
  • E 5 or 9

Q7:

What complex number lies at the midpoint of the complex numbers 𝑎 + 𝑏 𝑖 and 𝑥 + 𝑦 𝑖 , where 𝑎 , 𝑏 , 𝑥 , and 𝑦 are real, when they are represented on a complex plane?

  • A 𝑏 𝑦 2 + 𝑎 𝑥 2 𝑖
  • B ( 𝑎 𝑥 ) + ( 𝑏 𝑦 ) 𝑖
  • C 𝑎 𝑥 2 + 𝑏 𝑦 2 𝑖
  • D 𝑎 + 𝑥 2 + 𝑏 + 𝑦 2 𝑖
  • E ( 𝑎 + 𝑥 ) + ( 𝑏 + 𝑦 ) 𝑖

Q8:

What is the distance between the complex numbers 𝑎 + 𝑏 𝑖 and 𝑥 + 𝑦 𝑖 , where 𝑎 , 𝑏 , 𝑥 , and 𝑦 are real?

  • A ( 𝑎 + 𝑥 ) + ( 𝑏 + 𝑦 )
  • B ( 𝑎 + 𝑥 ) + ( 𝑏 + 𝑦 )
  • C ( 𝑎 𝑥 ) + ( 𝑏 𝑦 )
  • D ( 𝑎 𝑥 ) + ( 𝑏 𝑦 )
  • E ( 𝑎 𝑥 ) + ( 𝑏 𝑦 )

Q9:

What is the distance between the numbers 2 𝑖 and 2 + 𝑖 on the complex plane?

  • A 1
  • B 5
  • C2
  • D 4
  • E 3

Q10:

What is the distance between the numbers 2 + 3 𝑖 and 2 + 6 𝑖 on the complex plane?

  • A 3 5
  • B 9
  • C5
  • D 4
  • E 3

Q11:

What is the distance between the numbers 3 𝑖 and 7 𝑖 in the complex plane?

Q12:

What is the distance between the numbers 6 + 4 𝑖 and 3 𝑖 on the complex plane?

  • A 2 2
  • B 3 4
  • C 4
  • D 3
  • E 9

Q13:

Let 𝑧 and 𝑧 be points in the complex plane. A circle of center 𝑂 , where 𝑂 is the complex number 0 + 0 𝑖 , is plotted in the complex plane such that the line 𝑧 𝑧 is tangential to the circle at point 𝑄 . Point 𝑄 is a distance of 𝑘 | 𝑧 𝑧 | from 𝑧 . Given that 𝑧 = 8 1 9 𝑖 and 𝑧 = 7 + 𝑖 , find 𝑘 .

  • A 1 4
  • B 5 4
  • C 4 5
  • D 1 5
  • E 3 5

Q14:

What is the distance between the numbers 3 𝑖 and 7 𝑖 in the complex plane?

Q15:

What is the distance between the numbers 9 3 𝑖 and 4 + 4 𝑖 in the complex plane?

  • A 7 4
  • B 9 0
  • C12
  • D 1 7 0
  • E6

Q16:

What is the distance between the numbers 6 + 2 𝑖 and 1 + 2 𝑖 in the complex plane?

Q17:

Let 𝑧 , 𝑚 , and 𝑧 be complex numbers such that 𝑚 lies at the midpoint of the line segment connecting 𝑧 to 𝑧 . Given that 𝑧 = 4 + 5 𝑖 and 𝑚 = 1 2 + 2 0 𝑖 , find 𝑧 .

  • A 4 + 2 5 2 𝑖
  • B 2 0 1 0 𝑖
  • C 2 8 + 5 𝑖
  • D 2 8 + 3 5 𝑖
  • E 8 + 1 5 2 𝑖

Q18:

A triangle has its vertices at points 𝑎 , 𝑏 , and 𝑐 in the complex plane. Find an expression for the centroid of the triangle in terms of 𝑎 , 𝑏 , and 𝑐 . You can use the fact that the centroid divides the median in a ratio of 2 1 .

  • A 𝑎 + 𝑏 𝑐
  • B 𝑎 + 𝑏 + 𝑐 3
  • C 𝑎 + 𝑏 + 𝑐 2
  • D 𝑎 + 𝑏 + 𝑐 9
  • E 2 ( 𝑎 + 𝑏 + 𝑐 ) 3

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