Let $\vec{AB}=3\hat i+0\hat j+4\hat k$ and $\vec{AC}=5\hat i-2\hat j+4\hat k$.

The median through $A$ goes from vertex $A$ to the midpoint of side $BC$. If $M$ is the midpoint of $BC$, then $\vec{AM}=\frac{\vec{AB}+\vec{AC}}{2}.$

Compute $\vec{AB}+\vec{AC}$: $\vec{AB}+\vec{AC}=(3+5)\hat i+(0-2)\hat j+(4+4)\hat k=8\hat i-2\hat j+8\hat k.$

Hence, $\vec{AM}=\frac{1}{2}(8\hat i-2\hat j+8\hat k)=4\hat i-\hat j+4\hat k.$

Now find its magnitude:

Therefore, the length of the median through $A$ is $\boxed{\sqrt{33}}.$

Option check:

• A: $\sqrt{18}$ ❌
• B: $\sqrt{72}$ ❌
• C: $\sqrt{33}$ ✅
• D: $\sqrt{45}$ ❌