Solution:

The area with included angle $\theta$ is $36\sin\theta,$ so two triangles of equal area use angles $\theta$ and $180^\circ - \theta,$ with cosines $\pm\cos\theta.$ By the law of cosines the third sides satisfy $t^2 = 145 \mp 144\cos\theta,$ hence $t_1^2 + t_2^2 = 290.$ The only integer values in the valid range $(1, 17)$ are $11$ and $13$ ($121 + 169 = 290$), so the sum is $24.$

Thus, the correct answer is C.