Exercises: Calculate Distance and Midpoint on the Complex Plane

In the coordinate plane, what is the distance between the points $(1, 2)$ and $(4, 6)$?

What are the coordinates of the midpoint of the segment between $(2, 5)$ and $(6, 1)$ in the coordinate plane?

Compute the modulus $|z - w|$ where $z = 4 + 3i$ and $w = 1 + i$.

First compute $z - w$, then find its modulus. Enter the exact value.

Find the distance between $z = 5 + 2i$ and $w = 2 + 6i$.

Distance $= |z - w|$. Enter the exact simplified value.

Find the distance from $z = -1 + 4i$ to the origin (i.e., to $w = 0$).

This equals $|z - 0| = |z|$. Enter the exact value.

Find the midpoint $M$ of the segment from $z = 3 + 5i$ to $w = 7 + i$.

$M = \dfrac{z + w}{2} = $   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   $+$   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   $i$.

The midpoint of the segment from $z = 2 + 8i$ to $w = -4 + 2i$ is $M = $   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   $+$   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   $i$.

Find the distance between $z = 1 + i$ and $w = -2 + 5i$.

Enter the exact simplified value.