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battle of the 5 armies movie - SUpost

📅 July 7, 2026 👤 scraface

Jul 07, 2026

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📌 Question: A triangle has side lengths 13 cm, 14 cm, and 15 cm. Find the length of the shortest altitude in centimeters.

📌 Use Heron’s formula to compute the area. Semi-perimeter $ s $:

📌 Area $ A $:

📌 = \sqrt{21 \cdot 8 \cdot 42} = \sqrt{7056} = 84 \text{ cm}^2

📌 The shortest altitude corresponds to the longest side, which is 15 cm. Using area formula $ A = \frac{1}{2} \times \text{base} \times \text{height} $:

📌 = \frac{1}{2} \cdot 15 \cdot h \Rightarrow 168 = 15h \Rightarrow h = \frac{168}{15} = 11.2

📌 Thus, the shortest altitude is $ \boxed{11.2} $.Question: Find the vector $\mathbf{v}$ such that $\mathbf{v} \times \mathbf{b} = \mathbf{c} - \mathbf{b}$, where $\mathbf{b} = \langle 1, 2, 3 \rangle$ and $\mathbf{c} = \langle 4, 5, 6 \rangle$.

📌 Solution: Let $\mathbf{v} = \langle v_1, v_2, v_3 \rangle$. Compute $\mathbf{v} \times \mathbf{b}$:

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