Question

use the pythagorean theorem to derive the equation of the circle centered at the origin with a diameter of 6.
to which triangle should you apply the pythagorean theorem?
what is the equation of the circle?
$x^2 + y^2 = 9$
$(x + 3)^2 + (y + 3)^2 = 9$
$x^2 + y^2 = 36$
$(x + 3)^2 + (y + 3)^2 = 36$

Explanation:

Step1: Find circle radius

Radius $r = \frac{\text{Diameter}}{2} = \frac{6}{2} = 3$

Step2: Identify correct triangle

Use the right triangle with vertices $(0,0)$, $(x,0)$, $(x,y)$ (left diagram). The legs are $x$ and $y$, hypotenuse is radius $r$.

Step3: Apply Pythagorean theorem

$x^2 + y^2 = r^2$
Substitute $r=3$: $x^2 + y^2 = 3^2 = 9$

Answer:

1. The triangle with vertices $(0,0)$, $(x,0)$, and $(x,y)$ (the left diagram)
2. $x^2 + y^2 = 9$