Question

Points $z_1$ & $z_2$ are adjacent vertices of a regular octagon. The vertex $z_3$ adjacent to $z_2(z_3\ne z_1)$ can be represented by

• A. $z_3=\frac{1}{\sqrt{2}}(1\pm i)(z_1+z_2)$
• B. $z_3=\frac{1}{\sqrt{2}}(-1\pm i)(z_1-z_2)$
• C. $z_3=\frac{1}{\sqrt{2}}(-1\pm i)(z_2-z_1)$
• D. none of these

Answer 1

The correct option is B $z_3=\frac{1}{\sqrt{2}}(-1\pm i)(z_1-z_2)$

Points $z_1\xi z_2$ are adjacent vertices of a regular octagon. The vertex $z_3$ is adjacent to $z_2.$

We assume that $z_3\ne z_1$

As all sides of a regular octagon have the same length,

$\therefore |z_3-z_2|=|z_2-z_1|⟶\left(1\right)$

The angle between consecutive sides of a regular octagon is $\frac{\pi }{4}$

$\Rightarrow arg(z_3-z_2)=arg(z_2-z_1)\pm \frac{\pi }{4}⟶\left(2\right)$

From $\left(1\right)$ $\xi$ $\left(2\right)$ we can find an expression,

$\therefore z_3=\frac{1}{\sqrt{2}}(-1\pm i)(z_1-z_2)$

Hence, the answer is $z_3=\frac{1}{\sqrt{2}}(-1\pm i)(z_1-z_2).$