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can you get a headache from dehydration - Decision Point

📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 5Question: A soil scientist is analyzing a triangular plot of land with vertices at $(0, 0)$, $(6, 0)$, and $(3, 4)$. She needs to place a sensor at a point inside the triangle that minimizes the maximum distance to the three vertices. This point is known as the *Chebyshev center* of the triangle. Find the coordinates of this Chebyshev center.

📌 The Chebyshev center of a triangle is the center of the smallest circle that contains the triangle — also known as the **circumcenter** in the case of an acute triangle, or the point equidistant from the farthest vertices in degenerate cases. Since the triangle formed by $(0, 0)$, $(6, 0)$, and $(3, 4)$ is acute, the Chebyshev center coincides with the circumcenter.

📌 To find the circumcenter, we compute the perpendicular bisectors of two sides and find their intersection.

📌 Step 1: Compute midpoint and slope of side $AB$

📌 Let $A = (0, 0)$, $B = (6, 0)$.

📌 Midpoint of $AB$:

📌 M_1 = \left( \frac{0 + 6}{2}, \frac{0 + 0}{2} \right) = (3, 0)

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