Answer: C

Small Hint:

Draw the altitude from the top vertex to the base.

Big Hint:

In an isosceles triangle, that altitude splits the base into two equal parts.

Solution:

Construct $\overline{BD}$ as the altitude from $B$ to $\overline{AC}.$

Then $$60 = \dfrac{1}{2} \cdot BD \cdot 24,$$ which gives us that $BD = 5.$

From this, we apply the Pythagorean Theorem on $\triangle ABD:$ $$AB^2 = 5^2 + 12^2 = 169 = 13^2.$$ This gives us that $AB = 13.$

Thus, C is the correct answer.