Part 3/5:
When graphing, the upper semicircle is represented by the positive square root, while the lower semicircle is given by the negative square root. By evaluating specific values, we can ascertain the corresponding points on the circle. For instance, at ( x = a ), ( y ) will reach its maximum and minimum values of ( b + r ) and ( b - r ), respectively.
Confirming the Radius
To further establish that every point on the circle maintains a consistent distance ( r ) from the center ( (a, b) ), we would use the Pythagorean theorem. An arbitrary point ( (x, y) ) can serve as a part of a right-angled triangle with sides defined as ( \Delta x ) and ( \Delta y ):
( \Delta x = x - a )
( \Delta y = y - b )