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Three complex numbers z_1, z_2, z_3 are the vertices of an…
Question
Three complex numbers z_1, z_2, z_3 are the vertices of an equilateral triangle in the complex plane. One vertex is z_1 = 1 + 2i, and the centroid of the triangle is at 4 + 5i. Compute |z_2 - z_3|^2 (the square of a side length). Report the exact integer value.
✓ Verified answer: 54checked by our engine — not a guess
Step-by-step solution
For an equilateral triangle the centroid coincides with the circumcenter, and the circumradius is R = |z_1 - centroid|.
Here z_1 - centroid = (1 + 2i) - (4 + 5i) = -3 - 3i, so R^2 = 9 + 9 = 18.
The side s of an equilateral triangle relates to its circumradius by s = R*sqrt(3), hence s^2 = 3R^2 = 3*18 = 54.
Since all sides are equal, |z_2 - z_3|^2 = 54.
Final answer54
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