Video Transcript
Let 𝑧 one, 𝑚, and 𝑧 two be
complex numbers such that 𝑚 lies at the midpoint of the line segment connecting 𝑧
one to 𝑧 two. Given that 𝑧 two equals four plus
five 𝑖 and 𝑚 equals negative 12 plus 20𝑖, find 𝑧 one.
Well, we could draw an Argand
diagram and reason geometrically. But there’s another way. We know that the midpoint 𝑚 is the
arithmetic mean of the complex numbers 𝑧 one and 𝑧 two. And we can rearrange this equation
to find 𝑧 one in terms of 𝑚 and 𝑧 two. We multiply both sides by two,
subtract 𝑧 two from both sides, and swap the sides to find that 𝑧 one is two 𝑚
minus 𝑧 two. We know the values of 𝑚 and 𝑧
two. And so we substitute them. We distribute the two and the minus
sign and simplify to find that 𝑧 one is negative 28 plus 35𝑖.
