Coordinates

To graph it out, we would see a right triangle with legs of length $3\sqrt{5}$ and $d + 6$. If the hypotenuse is $3d$, then we can use the pythagorean theorem to get this equation:

$(3\sqrt{5})^2 + (d + 6)^2 = (3d)^2$

Simplified:

$45 + d^2 + 36 + 12d = 9d^2$

Combine like terms and subtraction:

$-8d^2 + 12d + 81 = 0$

Now we have a quadratic, we can apply the quadratic formula $d$ = $-b {+\over} \sqrt{b^2 - 4ac}\over2a$ where a is the coefficient of $d^2$, b is the coefficient of $d$, and c is the constant of the equation.

Plugging in the values, we get:

$-3 {+\over} 3\sqrt{19}\over-4$ = d

Since we need the smallest value of d, and d can't be a negative distance away from something, then we will use the subtraction operation. This simplifies to:

$d = {3 + 3\sqrt{19}\over4}$