The value of $\left(\sum_{k=1}^{n}\left(\sin\frac{2k\pi}{n} - i\cos\frac{2k\pi}{n}\right)\right)$ is (where $i$ is iota)
Step-by-Step Solution
Key Concept: The locus of $z$ is a circle; distances from the center to points on the circle are equal radii.
The perimeter is determined by finding the distances $CA$, $CB$, and $AB$. With $A = (4,2)$, $B = (10,0)$, we have $CA = CB = $ radius and $\angle ACB = 2\left(\frac{\pi}{3}\right) = \frac{\pi}{3}$. Computing $r^2 = (10-4)^2 + (0-2)^2 = 36 + 16 = 52$, so $r = \sqrt{26}$. Using the law of cosines or direct calculation, the triangle $ACB$ is isosceles with $CA = CB = \sqrt{26}$ and $AB = \sqrt{52}$. The perimeter is $\sqrt{26} + \sqrt{26} + \sqrt{52} = 2\sqrt{26} + 2\sqrt{26} = 4\sqrt{26}$ units.
Correct Answer: 4√26