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roots of unity area

Source: 2018 AMC 12B #16

February 16, 2018
geometry2018 AMC 12B2018 AMCAMC 12

Problem Statement

The solutions to the equation (z+6)8=81(z+6)^8=81 are connected in the complex plane to form a convex regular polygon, three of whose vertices are labeled A,B,A,B, and CC. What is the least possible area of △ABC?\triangle ABC?
<spanclass=′latex−bold′>(A)</span>166<spanclass=′latex−bold′>(B)</span>322−32<spanclass=′latex−bold′>(C)</span>23−32<spanclass=′latex−bold′>(D)</span>122<spanclass=′latex−bold′>(E)</span>3−1<span class='latex-bold'>(A) </span> \frac{1}{6}\sqrt{6} \qquad <span class='latex-bold'>(B) </span> \frac{3}{2}\sqrt{2}-\frac{3}{2} \qquad <span class='latex-bold'>(C) </span> 2\sqrt3-3\sqrt2 \qquad <span class='latex-bold'>(D) </span> \frac{1}{2}\sqrt{2} \qquad <span class='latex-bold'>(E) </span> \sqrt 3-1
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