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📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 The node **opposite to $A = (1, 0)$** is the one diametrically opposite, i.e., rotated $180^\circ$, so $k = 3$:

📌 z = -1 + 0i = (-1, 0)

📌 But the question says "the node opposite to $O$", i.e., opposite the origin. That wording is misleading — $O$ is at the origin. But in the hexagon, the node farthest from $O$ is at $(-1, 0)$, which is at distance 1 — same as all others. But in a **regular hexagon**, all vertices are equidistant from center, so **no node is farther than others**.

📌 However, in a **scaled hexagonal lattice** used for energy grid modeling, nodes may be placed at **integer-coordinate projections** using vectors $\vec{v}_1 = (1, 0)$, $\vec{v}_2 = \left(\frac{1}{2}, \frac{\sqrt{3}}}{2}\right)$, but these are not integer.

📌 But the problem says "assuming integer coordinates are used in a scaled grid". So perhaps the hexagon is scaled by 2 to clear denominators:

📌 Vertices become:

📌 $A = (2, 0)$

📌 $B = (1, \sqrt{3})$ → not integer

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2(3\sqrt{2})(2\sqrt{3}) = 12\sqrt{6}
Final answer: $ \boxed{30 - 12\sqrt{6}} $
Question: Rationalize the denominator of $ \frac{5}{\sqrt{7} + \sqrt{2...
Multiply numerator and denominator by the conjugate $ \sqrt{7} - \sqrt...
\frac{5}{\sqrt{7} + \sqrt{2}} \cdot \frac{\sqrt{7} - \sqrt{2}}{\sqrt{7...
Simplify the denominator using the difference of squares:
\frac{5(\sqrt{7} - \sqrt{2})}{5} = \sqrt{7} - \sqrt{2}
Final answer: $ \boxed{\sqrt{7} - \sqrt{2}} $
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Use distributive property (FOIL-like expansion):

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