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Home » The area is the magnitude of the cross product $\mathbfr \times \mathbfs$: - Portal da AcÃºstica

The area is the magnitude of the cross product $\mathbfr \times \mathbfs$: - Portal da AcÃºstica

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 A = \sqrt{s(s - a)(s - b)(s - c)} = \sqrt{21(21 - 13)(21 - 14)(21 - 15)} = \sqrt{21 \cdot 8 \cdot 7 \cdot 6} = \sqrt{7056} = 84

📌 The altitude corresponding to side $a$ is $h_a = \frac{2A}{a} = \frac{168}{13} \approx 12.92$, for $b$ it is $h_b = \frac{168}{14} = 12$, and for $c$ it is $h_c = \frac{168}{15} = 11.2$. The shortest altitude corresponds to the longest side, which is $15$, yielding:

📌 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:

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