First slide

Geometry of complex numbers

Question

Suppose $z_{1}, z_{2}, z_{3}$ are vertices of an equi-lateral triangle whose circumcentre $- 3 + 4 i,$ then $∣ z_{1} + z_{2} +$ $z_{3} ∣$ is equal to

Moderate

A

5
B

$10 \sqrt{3}$
C

15
D

$15 \sqrt{3}$

Solution

As triangle is equilateral, circumcentre and centroid of the triangle coincides, therefore,
$\begin{array}{l} \frac{1}{3} (z_{1} + z_{2} + z_{3}) = - 3 + 4 i \\ \Rightarrow |z_{1} + z_{2} + z_{3}| = 3 \sqrt{(- 3)^{3} + 4^{2}} = 15 \end{array}$

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