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Home » Try \( n = 1 \): \( 1^3 = 1 \equiv 1 \mod 8 \) - Veritas Home Health

Try \( n = 1 \): \( 1^3 = 1 \equiv 1 \mod 8 \) - Veritas Home Health

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 + 2 = \boxed{9}

📌 Question: A regular tetrahedron has three of its vertices at $(0,0,0)$, $(1,1,0)$, and $(1,0,1)$. Find the coordinates of the fourth vertex, assuming all coordinates are integers.

📌 Let the fourth vertex be $(x, y, z)$, and recall that in a regular tetrahedron, all edges have equal length.

📌 Compute edge lengths between the known points:

📌 \text{Distance between } (0,0,0) \text{ and } (1,1,0): \sqrt{(1)^2 + (1)^2 + 0} = \sqrt{2} \\

📌 \text{Distance between } (0,0,0) \text{ and } (1,0,1): \sqrt{(1)^2 + 0 + (1)^2} = \sqrt{2} \\

📌 \text{Distance between } (1,1,0) \text{ and } (1,0,1): \sqrt{0 + (1)^2 + (1)^2} = \sqrt{2}

📌 So all edges are of length $\sqrt{2}$.

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