Find the only positive real number $x$ for which $\displaystyle \frac{x-4}{9} = \frac{4}{x-9}$.
To find the positive real number $x$ that satisfies the equation, I need to isolate $x$ on one side of the equation and simplify the expression on the other side. To do that, I will first multiply both sides of the equation by $9(x-9)$, which is the product of the denominators of the fractions. This will eliminate the fractions and leave me with a quadratic equation. Multiplying both sides by $9(x-9)$, I get $(x-4)(x-9) = 36$. I can factor out $x$ from the left side and get $x(x-13) = 0$. Using the zero product property, I can set each factor equal to zero and get $x = 0$ or $x = 13$. However, $x = 0$ is not a positive real number, so I can eliminate that solution. Therefore, the only positive real number that satisfies the equation is $x = 13$. 