A circle with center at (2, -3) passes through the point (5, 1). What is the length of the radius?
["Determining the Radius of a Circle: A Practical Guide with Center and Point", "When working with circles in coordinate geometry, one of the most fundamental tasks is calculating the radius. In this article, we explore a specific case involving a circle with a known center and a point lying on its perimeter—and how to find the radius using the distance formula.", "---", "### Understanding the Circle Equation", "A circle is defined as the set of all points that are equidistant from a fixed point called the center. The standard equation of a circle with center ((h, k)) and radius (r) is:", "[\n(x - h)^2 + (y - k)^2 = r^2\n]", "Given the center at ((2, -3)) and a point ((5, 1)) lying on the circle, the distance from the center to this point is exactly the radius (r).", "---", "### Step-by-Step: Calculating the Radius", "To find the radius, compute the distance between the center ((2, -3)) and the point ((5, 1)) using the distance formula:", "[\nr = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\n]", "Substitute the coordinates:", "[\nr = \sqrt{(5 - 2)^2 + (1 - (-3))^2}\n]", "Simplify the differences:", "[\nr = \sqrt{(3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5\n]", "---", "### The Result: Radius Equals 5 Units", "The radius of the circle centered at ((2, -3)) passing through ((5, 1)) is exactly 5 units.", "This explanation highlights why the distance formula is essential—transforming geometric relationships into precise numerical values. Understanding this process helps in solving real-world problems involving circles, from designing wheels to modeling trajectories.", "---", "Key Takeaway:\nTo find the radius of a circle from its center ((h, k)) and a point ((x, y)) on the circle, calculate:", "[\nr = \sqrt{(x - h)^2 + (y - k)^2}\n]", "For the given example, the radius is 5. This method applies universally across mathematics and engineering disciplines.", "---", "Keywords:\ncircle radius, circle from center and point, distance formula coordinates, coordinate geometry, radius calculation, geometry tips, math solution, Pythagorean theorem in circles."]
