Given angle at circumference = 30^.

The angle subtended by a chord at the center of a circle is twice the angle subtended by the same chord at any point on the circumference.
Given angle at circumference $= 30^\circ$.
Angle at center, $\theta = 2 \times 30^\circ = 60^\circ$.

The length of a chord $L$ in a circle with radius $r$ and subtending an angle $\theta$ at the center is given by the formula:

Given $L = 14 \text{ cm}$ and $\theta = 60^\circ$.

We know that $\sin(30^\circ) = \frac{1}{2}$.

The radius of the circle is 14 cm.

The final answer is $\boxed{\text{14 cm}}$.

:

Initial area $A_{initial} = 25 \text{ cm}^2$.
First transformation matrix $M_1 = \begin{pmatrix} 7 & 3 \\ 1 & -1 \end{pmatrix}$.
Second transformation matrix $M_2 = \begin{pmatrix} 2 & -3 \\ 0 & 4 \end{pmatrix}$.

The area scale factor for a transformation matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ is given by the absolute value of its determinant, $|ad - bc|$.
For $M_1$:

The area scale factor for $M_1$ is $|-10| = 10$.

For $M_2$:

The area scale factor for $M_2$ is $|8| = 8$.

When multiple transformations are applied sequentially, the total area scale factor is the product of the individual area scale factors.
Total area scale factor $= (\text{Area scale factor for } M_1) \times (\text{Area scale factor for } M_2)$
Total area scale factor $= 10 \times 8 = 80$.

The area of the final image is the initial area multiplied by the total area scale factor.

The area of the final image is 2000 cm².

The final answer is $\boxed{\text{2000 cm}^2}$.