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Home » $-3v_1 + 2v_3 = 5$ - jntua results

$-3v_1 + 2v_3 = 5$ - jntua results

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 Question: A glaciologist models a cross-section of a glacier as a right triangle, where the hypotenuse represents the slope surface of length $z$, and the inscribed circle has radius $c$. If the ratio of the area of the circle to the area of the triangle is $\frac{\pi c^2}{\frac{1}{2}ab}$, express this ratio in terms of $z$ and $c$.

📌 Solution: In a right triangle with legs $a$, $b$, and hypotenuse $z$, the inradius is given by:

📌 c = \frac{a + b - z}{2}

📌 The area of the triangle is $A = \frac{1}{2}ab$, and the area of the inscribed circle is $\pi c^2$. The ratio is:

📌 \frac{\pi c^2}{\frac{1}{2}ab}

📌 From the identity $a + b = 2c + z$, and using the Pythagorean theorem $a^2 + b^2 = z^2$, we aim to express $ab$ in terms of $z$ and $c$. Note:

📌 Thus, the ratio becomes:

📌 \frac{\pi c^2}{\frac{1}{2}(2c^2 + 2cz)} = \frac{\pi c^2}{c^2 + cz} = \frac{\pi c}{c + z}

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