Distance and midpoint in the complex plane

About this lesson

Precalculus › Parametric equations and polar coordinates

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Distance and midpoint in the complex plane

Warm-up

Three quick questions from earlier lessons.

Discover

The point 3 + 4i is plotted at (3, 4). How far is it from 0?

  1. 5
  2. 7

Key idea

Distance is the modulus of a difference; the midpoint is an average.

Plot a+bia + bi at (a,b)(a, b). The modulus ∣a+bi∣=a2+b2|a + bi| = \sqrt{a^2 + b^2} is its distance from 0. The distance between zz and ww is ∣z−w∣|z - w|, and their midpoint is z+w2\dfrac{z + w}{2}.

Worked example

z = 1 + 2i, w = 4 − 2i.

z − w = −3 + 4i
Difference.
|−3 + 4i| = 5
Distance.
(5 + 0i)/2 = 2.5
Midpoint.

Your turn, guided

Your turn: |6 + 8i|.

1010
√(36 + 64).

Practice ★

  1. What is the modulus |7 + 8i|, the distance from 7 + 8i to 0?
  2. What is the modulus |2 + 5i|, the distance from 2 + 5i to 0?
  1. What is the real part of the midpoint of 5 + -7i and -2 + -7i?

Watch out!

Practice ★★

  1. What is the distance between 4 + 3i and -4 + -6i in the complex plane?
  2. What is the distance between -6 + 3i and 5 + 0i in the complex plane?
  1. What is the real part of the midpoint of 5 + -7i and -2 + -7i?

Exit ticket

Three questions to check you have got it.

  1. What is the distance between 4 + 3i and -4 + -6i in the complex plane?
  2. What is the distance between -6 + 3i and 5 + 0i in the complex plane?
  3. What is the distance between 3 + 3i and 1 + -3i in the complex plane?

Summary card

Distance and midpoint in the complex plane

|z − w| and (z + w)/2.

Watch out: distance uses √(x² + y²), not x + y.

End of the lesson

Back to the chapter

Answers: Distance and midpoint in the complex plane

A

  1. 113\sqrt{113}
  2. 29\sqrt{29}

B

  1. 32\frac{3}{2}

C

  1. 145\sqrt{145}
  2. 130\sqrt{130}

D

  1. 32\frac{3}{2}

E

  1. 145\sqrt{145}
  2. 130\sqrt{130}
  3. 2102 \sqrt{10}

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