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The radius is 6 units. - AssociationVoting

📅 May 16, 2026 👤 scraface

May 16, 2026

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📌 5Question: Compute $\cos 150^\circ$ if the angle is formed by the diagonal of a square divided by its side length in a cartographic map projection.

📌 Solution: The angle between the diagonal of a square and its side is $45^\circ$, but here the angle is $150^\circ$, which lies in the second quadrant. Using the reference angle $30^\circ$, $\cos 150^\circ = -\cos 30^\circ = -\frac{\sqrt{3}}{2}$. $\boxed{-\dfrac{\sqrt{3}}{2}}$

📌 Question: A linguist models language divergence as a triangular lattice with sides of lengths 9, 12, and 15 units. Find the length of the shortest altitude.

📌 Solution: The triangle with sides 9, 12, 15 is right-angled (since $9^2 + 12^2 = 15^2$). The area is $\frac{1}{2} \cdot 9 \cdot 12 = 54$. The altitudes correspond to $\frac{2 \cdot \text{Area}}{\text{side}}$, so:

📌 Altitude to hypotenuse (15): $\frac{108}{15} = 7.2$

📌 Altitude to 12: $\frac{108}{12} = 9$

📌 Altitude to 9: $\frac{108}{9} = 12$

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