Derive the equation of a circle from center/radius and complete the square to identify center/radius.
Write a circle equation from the endpoints of a diameter
Problem
A circle has diameter endpoints \((0,~0)\) and \((6,~8)\). Find the midpoint for the center, calculate the endpoint distance and halve it for the radius, then write the center-radius equation.
Diameter endpoints determine the circle in two different ways: their midpoint locates the center, while their separation determines twice the radius. We’ll average coordinates, compute the full endpoint distance, halve that distance, place the center shifts and squared radius into standard form, and test both endpoints against the equation.
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