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Home » But given the context, if $ u $ - IX Labs

But given the context, if $ u $ - IX Labs

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 Question:** A biologist studying the genetic diversity of plants is analyzing a regular tetrahedron with vertices at \((0, 0, 0)\), \((1, 0, 0)\), \((0, 1, 0)\), and a fourth vertex \((x, y, z)\) where \(x, y, z\) are integers. Find the coordinates of the fourth vertex.

📌 A regular tetrahedron has equal edge lengths. The given vertices are \((0, 0, 0)\), \((1, 0, 0)\), \((0, 1, 0)\). The side length is 1.

📌 The distance from \((0, 0, 0)\) to \((x, y, z)\) is:

📌 \sqrt{x^2 + y^2 + z^2} = 1 \quad \Rightarrow \quad x^2 + y^2 + z^2 = 1

📌 The distance from \((1, 0, 0)\) to \((x, y, z)\) is:

📌 \sqrt{(x-1)^2 + y^2 + z^2} = 1 \quad \Rightarrow \quad (x-1)^2 + y^2 + z^2 = 1

📌 x^2 - 2x + 1 + y^2 + z^2 = 1 \quad \Rightarrow \quad x^2 + y^2 + z^2 - 2x + 1 = 1

📌 Since \(x^2 + y^2 + z^2 = 1\), we have:

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