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+5𝑖 and 713𝑖.

  • A4+5𝑖
  • B2+9𝑖
  • C2+9𝑖
  • D54𝑖
  • E108𝑖

Q2:

A complex number 𝑤 lies at a distance of 52 from 𝑧=3+5𝑖 and at a distance of 45 from 𝑧=62𝑖. 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?

  • A𝑧+𝑧2
  • B𝑧𝑧2
  • C𝑧+𝑧2
  • D2(𝑧𝑧)
  • E𝑧𝑧2

Q4:

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

  • A2+4𝑖
  • B1+2𝑖
  • C8+10𝑖
  • D4+5𝑖
  • E2+𝑖

Q5:

Find the complex number 𝑧 such that 4+3𝑖 lies at the midpoint of 𝑧 and 34𝑖 when they are represented on a complex plane.

  • A11+2𝑖
  • B5+10𝑖
  • C5+13𝑖
  • D1+7𝑖
  • E7𝑖

Q6:

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

  • A3 or 11
  • B1.7 or 15.7
  • C5
  • D5 or 9
  • E0.5 or 14.5

Q7:

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

  • A𝑏𝑦2+𝑎𝑥2𝑖
  • B𝑎+𝑥2+𝑏+𝑦2𝑖
  • C𝑎𝑥2+𝑏𝑦2𝑖
  • D(𝑎𝑥)+(𝑏𝑦)𝑖
  • E(𝑎+𝑥)+(𝑏+𝑦)𝑖

Q8:

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

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

Q9:

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

  • A1
  • B2
  • C3
  • D4
  • E5

Q10:

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

  • A35
  • B3
  • C4
  • D9
  • E5

Q11:

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

Q12:

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

  • A4
  • B3
  • C9
  • D22
  • E34

Q13:

Let 𝑧 and 𝑧 be points in the complex plane. A circle of center 𝑂, where 𝑂 is the complex number 0+0𝑖, is plotted in the complex plane such that the line 𝑧𝑧 is tangential to the circle at point 𝑄. Point 𝑄 is a distance of 𝑘|𝑧𝑧| from 𝑧. Given that 𝑧=819𝑖 and 𝑧=7+𝑖, find 𝑘.

  • A54
  • B45
  • C14
  • D15
  • E35

Q14:

What is the distance between the numbers 93𝑖 and 4+4𝑖 in the complex plane?

  • A170
  • B90
  • C12
  • D6
  • E74

Q15:

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

Q16:

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+5𝑖 and 𝑚=12+20𝑖, find 𝑧.

  • A28+5𝑖
  • B4+252𝑖
  • C2010𝑖
  • D8+152𝑖
  • E28+35𝑖

Q17:

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 21.

  • A𝑎+𝑏+𝑐2
  • B𝑎+𝑏+𝑐9
  • C𝑎+𝑏+𝑐3
  • D𝑎+𝑏𝑐
  • E2(𝑎+𝑏+𝑐)3

Q18:

Find the distance between the complex numbers 𝑧 and 𝑧 shown on the complex plane. Give your answer in an exact simplified form.

  • A4
  • B32
  • C6
  • D241
  • E8

Q19:

Find the distance between the complex numbers 𝑧 and 𝑧 shown on the complex plane. Give your answer in simplified exact form.

  • A55
  • B15
  • C113
  • D53
  • E313

Q20:

If 𝑧=𝑎+5𝑖, 𝑧=36𝑖, and |𝑧𝑧|=125, what are the possible values of 𝑎?

  • A5, 7
  • B1, 5
  • C1, 7
  • D1, 1
  • E5, 5

Q21:

Find the distance between the complex numbers 𝑧 and 𝑧 shown on the complex plane.

Give your answer in simplified exact form.

  • A15
  • B61
  • C313
  • D913
  • E113

Q22:

Let 𝑧, 𝑃, and 𝑧 be complex numbers such that 𝑃 lies at a point on the line 𝑧𝑧 that is 35 of the distance from 𝑧 to 𝑧. Given that 𝑧=311𝑖 and 𝑧=13+19𝑖, find 𝑃.

  • A7𝑖
  • B9+7𝑖
  • C97𝑖
  • D8+4𝑖
  • E48𝑖

Q23:

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

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