The standard equation of a circle with center $(h,k)$ and radius $r$ is $(x-h{)}^2+(y-k{)}^2=r^2$. Here, the center is $(3,-2)$, so $h=3$ and $k=-2$. Substituting gives $(x-3{)}^2+(y-(-2){)}^2=5^2$. Simplifying, $(x-3{)}^2+(y+2{)}^2=25$. Therefore, choice A must be true for every point on the circle.

Because the path goes from one point on the circle to the opposite point and passes through the center, the path is a diameter. The problem states that the diameter of the garden is 18 feet, so the length of the path is 18 feet.

The distance around a circle is its circumference, given by the formula $C=2\pi r$. The radius of the splash pad is 6 feet, so substitute $r=6$ into the formula: $C=2\pi (6)=12\pi$. Therefore, the correct answer is $12\pi$.

The equation of a circle with center $(h,k)$ is $(x-h{)}^2+(y-k{)}^2=r^2$. Since the center is $(3,-2)$, the equation must be $(x-3{)}^2+(y+2{)}^2=r^2$. To find $r^2$, use the point $(6,2)$ on the circle: horizontal change $=6-3=3$ and vertical change $=2-(-2)=4$. So the radius is $\sqrt{3^2+4^2}=\sqrt{9+16}=5$, which means $r^2=25$. Therefore, the equation is $(x-3{)}^2+(y+2{)}^2=25$.

The standard form of a circle is $(x-h{)}^2+(y-k{)}^2=r^2$, where $(h,k)$ is the center and $r$ is the radius. The center is 3 units right of the origin, so $h=3$. It is 2 units below the origin, so $k=-2$. The radius is 5, so $r^2=25$. Substituting gives $(x-3{)}^2+(y-(-2){)}^2=25$, which simplifies to $(x-3{)}^2+(y+2{)}^2=25$.