Step 1: Use the cross product formula:
The distance of a point $P$ from a line through point $A$ with direction $\vec{b}$ is $\dfrac{|\overrightarrow{AP} \times \vec{b}|}{|\vec{b}|}$.
Take $A = (0, 0, 0)$ on the line and $\vec{b} = \hat{i}$. Then $\overrightarrow{AP} = 2\hat{i} + 5\hat{j} + 7\hat{k}$.
Step 2: Compute the cross product:
\[ \overrightarrow{AP} \times \hat{i} = (2\hat{i} + 5\hat{j} + 7\hat{k}) \times \hat{i} = 5(\hat{j} \times \hat{i}) + 7(\hat{k} \times \hat{i}) \]
We know $\hat{j} \times \hat{i} = -\hat{k}$ and $\hat{k} \times \hat{i} = \hat{j}$. So the result is $7\hat{j} - 5\hat{k}$.
Step 3: Take the magnitude:
$|7\hat{j} - 5\hat{k}| = \sqrt{49 + 25} = \sqrt{74}$, and $|\vec{b}| = 1$.
So the distance is $\sqrt{74}$.
Step 4: Quick sense check:
The line is the x-axis. The distance of a point from the x-axis is $\sqrt{y^2 + z^2} = \sqrt{25 + 49} = \sqrt{74}$. This agrees.
Options (A), (B) and (D) do not match this value.
Final Answer:
The perpendicular distance is $\sqrt{74}$, option (C).
\[ \boxed{\sqrt{74}} \]