Question

Use the picture and explain how the equation of a circle and the Pythagorean Theorem (a2 + b2 = c2) are related. Please be specific. Which "parts" are the same, etc?

Answer 1

That's a very important thing to know. 
The equation for the circle, if you put the circle at the origin, is x^2 + y^2 = r^2 which is the radius.

That is exactly the same thing as the pythagorean theorem. 

Now when you shift the circle so that it has a new center, you are using the distance formula to describe the radius. Notice the radius in the formula is unchanged no matter where a and b are. Wherever they are the radius will follow. You've just shifted your location. That's all. The distance formula just takes into account where the two points are (x,y) and (a,b) where a,b is the center. I say again, the distance formula really is pythagoras. And the circle is really pythagoras.

It's a special application of the pythagorean theorem. The Greeks, who developed a^2 + b^2 = c^2, knew nothing about a grid. It took almost 2000 years for that to become clear. 

The darkened bold part is the answer. It is the connection between the circle and pythagoras. The bridge from one to the other is the distance formula.