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Home » V_{\textsphere} = \frac43 \pi r^3 - Anadea

V_{\textsphere} = \frac43 \pi r^3 - Anadea

📅 May 15, 2026 👤 scraface

May 15, 2026

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📌 x + y = a \quad \text{و} \quad y + z = a

📌 نعوض $y = a - x$ في $y + z = a$:

📌 a - x + z = a \quad \Rightarrow \quad z = x

📌 نعوض $z = x$ و $y = a - x$ في $x^2 + y^2 + z^2 = 2a^2$:

📌 x^2 + (a - x)^2 + x^2 = 2a^2 \quad \Rightarrow \quad x^2 + a^2 - 2ax + x^2 + x^2 = 2a^2 \quad \Rightarrow \quad 3x^2 - 2ax + a^2 = 2a^2 \quad \Rightarrow \quad 3x^2 - 2ax - a^2 = 0

📌 x = \frac{2a \pm \sqrt{4a^2 + 12a^2}}{6} = \frac{2a \pm 4a}{6} \quad \Rightarrow \quad x = a \quad \text{أو} \quad x = -\frac{a}{3}

📌 بما أن $x$ يجب أن يكون عددًا صحيحًا، فإن $x = a$، وبالتالي $y = 0$ و $z = a$. لذلك، إحداثيات الرأس الرابع هي:

📌 \boxed{(a, 0, a)}

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