To make the left-hand side of the equation \(x^2 - 14x + c\) a perfect square trinomial, we need to determine the value of \(c\).
A perfect square trinomial can be expressed in the form of \((x - a)^2\), which expands to \(x^2 - 2ax + a^2\). Here, the coefficient for \(x\) is \(-14\), which corresponds to \(-2a\). From this, we can find \(a\):
\[ -2a = -14 \implies a = \frac{14}{2} = 7 \]
The constant term in the perfect square trinomial is \(a^2\):
\[ a^2 = 7^2 = 49 \]
So, we need \(c\) to equal \(49\) in order for \(x^2 - 14x + c\) to be a perfect square trinomial.
Therefore, the value of \(c\) that would make the left-hand side a perfect square trinomial is
\[ \boxed{49} \]