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.

["Finding the Chebyshev Center of a Triangle: A Soil Scientist’s Optimal Sensor Placement", "In landscape-scale environmental monitoring, placing sensors optimally within irregular plots can significantly improve data accuracy and efficiency. For a triangular parcel of land—especially in precision agriculture or soil analysis—identifying the Chebyshev center ensures the sensor minimizes the maximum distance to all three vertices. This point, equivalent to the center of the smallest enclosing circle (or the minimax center), is a key geometric concept in operations research and spatial planning.", "In this article, we analyze triangle $ABC$ with vertices at $A = (0, 0)$, $B = (6, 0)$, and $C = (3, 4)$, and determine the coordinates of its Chebyshev center—the point that minimizes the maximum distance to these three vertices.", "---", "### Understanding the Chebyshev Center", "The Chebyshev center of a convex polygon is the center of the smallest circle that contains the entire polygon. For a triangle, this point lies inside the triangle (or on its boundary in degenerate cases) and ensures that no vertex lies more than a fixed radius away. For a general triangle, the Chebyshev center coincides with the circumcenter if the triangle is acute; however, for obtuse triangles, it lies at the point minimizing the maximum distance to the vertices—often not the circumcenter.", "For any triangle, this point is the unique point $P$ such that the maximum distance from $P$ to $A$, $B$, and $C$ is minimized. This is equivalent to finding the center of the smallest enclosing disk.", "---", "### Step 1: Analyze Triangle Geometry", "Given:\n- $A = (0, 0)$\n- $B = (6, 0)$\n- $C = (3, 4)$", "First, compute side lengths to determine the triangle’s type:", "- $AB = \sqrt{(6 - 0)^2 + (0 - 0)^2} = 6$\n- $AC = \sqrt{(3 - 0)^2 + (4 - 0)^2} = \sqrt{9 + 16} = 5$\n- $BC = \sqrt{(3 - 6)^2 + (4 - 0)^2} = \sqrt{9 + 16} = 5$", "So triangle $ABC$ is isosceles with $AC = BC = 5$, $AB = 6$. Since $AC^2 + BC^2 = 25 + 25 = 50 > 36 = AB^2$, the triangle is acute. Therefore, its circumcenter lies inside the triangle, and the Chebyshev center coincides with the circumcenter.", "---", "### Step 2: Find the Circumcenter", "The circumcenter is the intersection of the perpendicular bisectors of two sides.", "Midpoint and perpendicular bisector of $AB$:\n- Midpoint of $AB$: $M_{AB} = \left(\frac{0+6}{2}, \frac{0+0}{2}\right) = (3, 0)$\n- Since $AB$ lies on the x-axis, its perpendicular bisector is the vertical line $x = 3$.", "Midpoint and perpendicular bisector of $AC$:\n- Midpoint of $AC$: $M_{AC} = \left(\frac{0+3}{2}, \frac{0+4}{2}\right) = (1.5, 2)$\n- Slope of $AC$: $\frac{4 - 0}{3 - 0} = \frac{4}{3}$, so the perpendicular slope is $-\frac{3}{4}$", "Equation of perpendicular bisector of $AC$:\n$$\ny - 2 = -\frac{3}{4}(x - 1.5)\n$$", "Substitute $x = 3$ (from the first bisector):\n$$\ny - 2 = -\frac{3}{4}(3 - 1.5) = -\frac{3}{4}(1.5) = -\frac{3}{4} \cdot \frac{3}{2} = -\frac{9}{8}\n$$\n$$\ny = 2 - \frac{9}{8} = \frac{16}{8} - \frac{9}{8} = \frac{7}{8}\n$$", "Thus, the circumcenter is at $(3, \frac{7}{8})$.", "---", "### Step 3: Verify It Minimizes the Maximum Distance", "Let $P = (3, \frac{7}{8})$. Compute distances to $A$, $B$, and $C$:", "- $PA = \sqrt{(3 - 0)^2 + \left(\frac{7}{8} - 0\right)^2} = \sqrt{9 + \frac{49}{64}} = \sqrt{\frac{576 + 49}{64}} = \sqrt{\frac{625}{64}} = \frac{25}{8} = 3.125$", "- $PB = \sqrt{(3 - 6)^2 + \left(\frac{7}{8} - 0\right)^2} = \sqrt{9 + \frac{49}{64}} = \frac{25}{8}$", "- $PC = \sqrt{(3 - 3)^2 + \left(\frac{7}{8} - 4\right)^2} = \sqrt{0 + \left(-\frac{25}{8}\right)^2} = \frac{25}{8}$", "All three distances are equal—$PA = PB = PC = \frac{25}{8}$. Hence, $P$ minimizes the maximum distance to the vertices.", "---", "### Conclusion", "The Chebyshev center of triangle $ABC$ is at $(3, \frac{7}{8})$, which lies at the circumcenter. This point ensures the most balanced sensor coverage—ideal for soil moisture or nutrient sensors in agricultural plots where uniform signal strength and minimal blind zones are critical.", "For soil scientists and land managers working with irregular plots, computing the Chebyshev center provides a mathematically elegant solution to optimal monitoring design.", "---", "Final Answer:\nThe Chebyshev center of the triangle is $\boxed{\left(3, \dfrac{7}{8}\right)}$."]