Problem:
The solutions to the equation (z+6)8=81 are connected in the complex plane to form a convex regular polygon, three of whose vertices are labeled A,B, and C. What is the least possible area of △ABC ?
Answer Choices:
A. 616
B. 232−23
C. 23−22
D. 212
E. 3−1 Solution:
The answer would be the same if the equation were z8=81, resulting from a horizontal translation of 6 units. The solutions to this equation are the 8 eighth roots of 81 , each of which is 834=3 units from the origin. These 8 points form a regular octagon. The triangle of minimum area occurs when the vertices of the triangle are consecutive vertices of the octagon, so without loss of generality they have coordinates A(216,216),B(3,0), and C(216,−216). This triangle has base AC=6 and height 3−216, so its area is
21⋅6⋅(3−216)=232−23
OR
The complex solutions form a regular octagon centered at z=−6. The distance from the center to any one of the vertices is 881= 834=3. By the Law of Cosines, the side length s of the octagon satisfies
s2=(3)2+(3)2−2⋅3⋅3⋅cos45∘=6−6⋅22=6−32.
The least possible area of △ABC occurs when two of the sides of △ABC are adjacent sides of the octagon; the angle between these two sides is 135∘. The sine formula for area gives