Phasor analysis of AC circuits: Which is correct: impedance triangle or phasors as resistors?

Phasor analysis is convenient with AC circuits when looking for simple quantities like impedance. After converting capacitive and inductive quantities to reactances, I then treat them like resistors. It's a convenient way to make AC circuit analysis relatively simple. However, I found out today that I may have been doing it wrong, and I need help understanding why. For example, the circuit below: To find the impedance in this circuit, I would first find XL, then Xc:

Then I would treat the reactances as resistances, in this case using the simple parallel resistor formula:

\begin{equation*} Z = \frac{1}{\frac{1}{R}+\frac{1}{X_L}+\frac{1}{X_C}} \end{equation*}

The result I get this way is Z = 24.75Ω

In studying for the FE exam, I've come across a method which relies on the impedance triangle:

Through some trig on the relations on this diagram, the formula for impedance in a parallel AC circuit with reactive components becomes:

\begin{equation*} Z = \frac{1}{ \sqrt{\left(\frac{1}{R}\right)^2+\left(\frac{1}{X_L}+\frac{1}{X_C}\right)^2} } \end{equation*}

The result I get this way is Z = 24.17Ω

As you can see, they both produce a very similar result, but not the same. One is an approximation, and I have a suspicion based on the impedance triangle that the latter method is correct.

Can someone please elaborate as to why that is or isn't the case?

Thank you.