Which is the standard equation for a circle centered at the origin with radius r?

The standard equation for a circle centered at the origin with radius r is:

x² + y² = r²

Here's the step-by-step explanation:

1. A circle is a set of points in a plane that are all the same distance from a fixed point, called the center of the circle.

2. When a circle is centered at the origin (0,0), any point (x,y) on the circle is a distance r from the origin.

3. By the Pythagorean theorem, the distance from the origin to the point (x,y) is √(x² + y²).

4. Therefore, for a point to be on the circle, we must have √(x² + y²) = r.

5. Squaring both sides of the equation to get rid of the square root gives us x² + y² = r².

So, x² + y² = r² is the standard equation for a circle centered at the origin with radius r.