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

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

Let 𝑧₁, π‘š, and 𝑧₂ be complex numbers such that π‘š lies at the midpoint of the line segment connecting 𝑧₁ to 𝑧₂. Given that 𝑧₂ = 4 + 5𝑖 and π‘š = βˆ’12 + 20𝑖, find 𝑧₁.

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Video Transcript

Let 𝑧 one, π‘š, and 𝑧 two be complex numbers such that π‘š lies at the midpoint of the line segment connecting 𝑧 one to 𝑧 two. Given that 𝑧 two equals four plus five 𝑖 and π‘š 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 π‘š 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 π‘š 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 π‘š minus 𝑧 two. We know the values of π‘š 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.

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