Problems involving similar triangles and prism volume calculations.

Since $AB$ is parallel to $CD$, triangles $\triangle ATB$ and $\triangle DTC$ are similar.
This means that the ratio of corresponding sides is equal.

We are given $AT : TD = 1 : 3$, so $\frac{AT}{TD} = \frac{1}{3}$.
Therefore, $\frac{BT}{TC} = \frac{1}{3}$.
This implies $TC = 3 \times BT$.

We are also given that $CB = 12$ cm.
We know that $CB = BT + TC$.
Substitute $TC = 3 \times BT$ into the equation:

The length of $TB$ is $\boxed{\text{3 cm}}$.

:

The formula for the area of an equilateral triangle with side $s$ is $A = \frac{\sqrt{3}}{4}s^2$.
Given side $s = 3$ cm.

The volume of a prism is given by the formula $V = \text{Area of cross-section} \times \text{Length}$.
Given length $= 25$ cm.

As a decimal, $V \approx \frac{225 \times 1.73205}{4} \approx \frac{389.71125}{4} \approx 97.4278 \text{ cm}^3$.
The volume of the prism is $\boxed{\frac{225\sqrt{3}}{4} \text{ cm}^3 \text{ or approximately } 97.43 \text{ cm}^3}$.

:

We need to find the greatest common divisor (GCD) of 36, 84, and 90.

The common prime factors are 2 and 3.
The lowest power of 2 present in all factorizations is $2^1$.
The lowest power of 3 present in all factorizations is $3^1$.

The capacity of the greatest vessel is $\boxed{6 \text{ litres}}$.