Question

derive the equation of the circle with a center at $(h,k)$ and a radius of $r$ using the pythagorean theorem.

Explanation:

Step1: Determine the lengths of the right - triangle sides

The horizontal side length of the right - triangle is \(|x - h|\), and the vertical side length is \(|y - k|\). The hypotenuse length is \(r\) (the radius of the circle).

Step2: Apply the Pythagorean theorem

According to the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(a = |x - h|\), \(b = |y - k|\), and \(c = r\). Substituting these values into the formula, we get \((x - h)^{2}+(y - k)^{2}=r^{2}\)

Answer:

\((x - h)^{2}+(y - k)^{2}=r^{2}\)