1. The problem asks which equation is used to derive the equation of a circle. 2. The standard equation of a circle with center at the origin is derived from the Pythagorean theorem, which states that for a right triangle with legs $a$ and $b$ and hypotenuse $r$, the relationship is: $$a^2 + b^2 = r^2$$ 3. In the context of a circle, $a$ and $b$ correspond to the $x$ and $y$ coordinates of any point on the circle, and $r$ is the radius. 4. Therefore, the equation of the circle is: $$x^2 + y^2 = r^2$$ 5. Among the options given for Question 1, the correct choice is: D. $x^2 + y^2 = r^2$ 6. For Question 2, the problem asks to choose two expressions that can be substituted for the variables on the left side of the equation to complete the derivation. 7. The general form of the circle equation when the center is at $(h, k)$ is: $$(x - h)^2 + (y - k)^2 = r^2$$ 8. From the options, the expressions that represent $(x - h)$ and $(y - k)$ are: - $x - 4$ (option C) - $y - 5$ (option G) 9. These correspond to a circle centered at $(4, 5)$. Final answers: - Question 1: D - Question 2: C and G