Step-by-step Solution:

To find the measure of an angle in a triangle when all three side lengths are known, we use the Law of Cosines.

1. State the Law of Cosines formula:
The Law of Cosines formula arranged to solve for angle C is:
\[\cos(C) = \frac{a^2 + b^2 - c^2}{2ab}\]
2. Substitute the given side lengths into the formula:
We are given the side lengths \(a=9\), \(b=7\), and \(c=12\).
\[\cos(C) = \frac{9^2 + 7^2 - 12^2}{2(9)(7)}\]
3. Calculate the value of \(\cos(C)\):
First, evaluate the squares of the side lengths.
\[\begin{aligned} \cos(C) &= \frac{81 + 49 - 144}{126} \\ &= \frac{130 - 144}{126} \\ &= \frac{-14}{126} \end{aligned}\]
Now, simplify the fraction by dividing the numerator and denominator by their greatest common divisor, 14.
\[\cos(C) = -\frac{1}{9}\]
4. Find the angle C:
To find the angle C, we take the inverse cosine (arccosine) of \(-\frac{1}{9}\). Since the cosine value is negative, the angle will be obtuse (greater than 90°).
\[C = \arccos\left(-\frac{1}{9}\right)\]
Using a calculator, this gives:
\[C \approx 96.38^\circ\]
5. Analysis of Options:
The calculated value for angle C is approximately \(96.38^\circ\). This value does not exactly match any of the provided options. The closest choice is (D) \(97.57^\circ\). This small discrepancy suggests there may be a typo in the side lengths or the options in the original question.

Result:
Based on a direct and correct application of the Law of Cosines, the angle C is approximately \(96.38^\circ\). The closest given answer choice is \(97.57^\circ\).