Solution: Let the radius of the circular lake be $ r $ km. The chord length is 10 km, so half the chord length is $ 5 $ km. The perpendicular distance from the center to the chord is 3 km. By the Pythagorean Theorem applied to the right triangle formed by the radius, half the chord, and the perpendicular from the center to the chord, we have:

["Title: How to Calculate the Radius of a Circular Lake Using Chord and Perpendicular Distance | Step-by-Step Solution", "---", "### Introduction\nUnderstanding the geometry of circular lakes or bodies of water is essential in fields like environmental science, urban planning, and land surveying. But what if you only know the chord length (a straight segment across the lake) and the perpendicular distance from the center to that chord? The good news is, with basic geometry—specifically the Pythagorean Theorem—you can easily determine the full radius of the circular lake.", "This article walks through a clear, step-by-step solution for finding the radius $ r $ (in kilometers) when given:\n- Chord length = 10 km\n- Half the chord length = 5 km\n- Perpendicular distance from the center to the chord = 3 km", "Let’s dive in!", "---", "### The Geometry Behind the Problem", "Imagine a perfect circle representing your circular lake. A chord cuts across it, like a bridge across a lake. The shortest distance from the center of the circle to the chord is a perpendicular line, forming the height of a right triangle. This setup allows us to apply the Pythagorean Theorem to find the radius.", "Let’s define the key parts of our right triangle:\n- One leg = half the chord length = $ 5 $ km\n- The other leg = perpendicular distance from center to chord = $ 3 $ km\n- The hypotenuse = radius $ r $ km (what we want to find)", "---", "### Applying the Pythagorean Theorem", "The Pythagorean Theorem states:\n$$\na^2 + b^2 = c^2\n$$\nWhere $ a $ and $ b $ are the legs, and $ c $ is the hypotenuse.", "Plugging in the known values:\n$$\n5^2 + 3^2 = r^2\n$$\n$$\n25 + 9 = r^2\n$$\n$$\n34 = r^2\n$$", "Now take the square root of both sides to solve for $ r $:\n$$\nr = \sqrt{34}\n$$", "---", "### Final Answer", "The radius $ r $ of the circular lake is:\n$$\n\boxed{\sqrt{34} \ ext{ km}} \quad (\approx 5.83 \ ext{ km})\n$$", "This elegant solution proves how simple geometric principles can unlock complex real-world problems. Whether measuring lake sizes, designing water features, or analyzing natural landscapes, mastering chord and distance calculations enables precise, confident decision-making.", "---", "### Why This Matters", "Knowing the radius helps estimate lake area, understand water volume, plan around the shoreline, or assess ecological needs. With just a chord and perpendicular distance, you can fully define the circular boundary—no fancy equipment required. Next time you encounter such a scenario, apply the Pythagorean Theorem and make informed insights with confidence!", "---", "### Key Terms for SEO\ncircular lake radius calculation, Pythagorean theorem for lake chords, geometry lake measurement, circular water body radius, simplify geometry problems, water body geometry, radius from chord and perpendicular distance", "---", "Author Bio\nExperts in environmental geometry and land surveying, we help professionals and learners translate real-world shapes into measurable data using fundamental math principles. Start solving circular geometry problems today.", "---\nTags: circular lake radius, chord length formula, Pythagorean Theorem application, geometry applied science, lake measurement guide"]