Step 1: List the given data.
$|\bar{a}| = 5$, $|\bar{b}| = 13$, $|\bar{a}\times\bar{b}| = 25$, and the angle satisfies $\tfrac{\pi}{2} < \theta < \pi$. We want $\bar{a}\cdot\bar{b}$.
Step 2: Recall the two products.
Cross product size: $|\bar{a}\times\bar{b}| = |\bar{a}||\bar{b}|\sin\theta$. Dot product: $\bar{a}\cdot\bar{b} = |\bar{a}||\bar{b}|\cos\theta$.
Step 3: Find $\sin\theta$.
\[ 25 = 5 \times 13 \times \sin\theta = 65\sin\theta \;\Rightarrow\; \sin\theta = \frac{25}{65} = \frac{5}{13} \]
Step 4: Find $\cos\theta$ in size.
\[ \cos^2\theta = 1 - \sin^2\theta = 1 - \frac{25}{169} = \frac{144}{169} \;\Rightarrow\; \cos\theta = \pm\frac{12}{13} \]
Step 5: Pick the correct sign.
The angle is between $90^\circ$ and $180^\circ$, where cosine is negative. So $\cos\theta = -\dfrac{12}{13}$.
Step 6: Find the dot product.
\[ \bar{a}\cdot\bar{b} = 5 \times 13 \times \left(-\frac{12}{13}\right) = 5 \times (-12) = -60 \]
\[ \boxed{-60} \]