Question Video: Finding a Complex Number given Another Complex Number and the Midpoint between the Two Mathematics

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.