Let the circular garden have center $ O $ and radius $ r = 5 $ meters (since the diameter is 10 meters). The walking path is a chord $ AB $ of length 6 meters. We are to find the perpendicular distance from the center $ O $ to this chord.
["Optimizing Walkability: Finding the Distance from the Circle Center to a Garden Path", "In any well-planned circular garden, accessibility and aesthetics go hand in hand. This article explores a practical geometric scenario: determining the perpendicular distance from the center $ O $ of a circular garden with radius $ r = 5 $ meters to a walking path marked by a chord $ AB = 6 $ meters long. This measurement is crucial for designing smooth, safe paths while preserving the garden’s natural flow.", "---", "### Understanding the Geometry", "Let the circular garden be defined with center $ O $ and radius $ r = 5 $ meters. The walking path lies along chord $ AB $, measuring 6 meters. Since the diameter of the garden is $ 10 $ meters, the chord is safely within the circle—never exceeding the boundary.", "We aim to compute the shortest distance from the center $ O $ to the chord $ AB $, which is the perpendicular distance $ d $. This distance splits the chord into two equal segments, each $ 3 $ meters long, forming two right triangles inside the circle.", "---", "### Applying the Pythagorean Theorem", "Let $ O $ be the center, and $ M $ the midpoint of chord $ AB $. Then $ OM \perp AB $, and triangle $ \ riangle OMA $ is a right triangle with:", "- Hypotenuse $ OA = r = 5 $ meters (the radius),\n- One leg $ AM = \frac{AB}{2} = \frac{6}{2} = 3 $ meters,\n- The other leg $ OM = d $, the distance from center to chord (to be found).", "By the Pythagorean Theorem:", "[\nOA^2 = OM^2 + AM^2\n]", "Substitute known values:", "[\n5^2 = d^2 + 3^2\n]", "[\n25 = d^2 + 9\n]", "[\nd^2 = 16\n]", "[\nd = \sqrt{16} = 4 \ ext{ meters}\n]", "---", "### Conclusion", "The perpendicular distance from the center $ O $ of the circular garden to the 6-meter walking path is 4 meters. This precise measurement ensures efficient path planning, maintains equal access on both sides, and supports the garden’s harmonious layout.", "Whether designing walking routes, placing benches, or installing lighting, knowing this geometric property allows landscape planners to optimize usability while respecting the elegant symmetry of circular spaces.", "---", "Keywords:\ncircular garden geometry, chord distance from center, perpendicular distance to chord, walking path radius 5 meters, chord length 6 meters, geometric optimization garden, center to chord distance, application geometry garden design", "---", "Manage your garden’s flow with math—ensure every step feels intentional."]
