Final answer:
Using the Pythagorean Theorem, the standard equation of a circle with center (a, b) and radius r is derived as (x - a)² + (y - b)² = r², where any point (x, y) lies on the circumference.
Step-by-step explanation:
To derive the standard equation of a circle from the Pythagorean Theorem, we begin by interpreting a circle as a set of all points that have a fixed distance, the radius, from a central point, typically denoted as (a, b). We consider a point (x, y) on the circumference of the circle.
By definition of a circle, the distance from this point to the center (a, b) is the radius, r. Let's call the horizontal leg of the right triangle formed the difference in x values (x - a) and the vertical leg the difference in y values (y - b). The Pythagorean Theorem states that a right triangle has sides of lengths a and b, and a hypotenuse of length c, with the relationship a² + b² = c².
In the context of the circle, the Pythagorean Theorem tells us that the sum of the squares of the legs equals the square of the hypotenuse, which is the radius of the circle squared:
(x - a)² + (y - b)² = r².
This equation is the standard form of the equation of a circle with center (a, b) and radius r. Understanding this relationship helps in analyzing the geometric properties of a circle and is fundamental in coordinate geometry.