An arc of length $121\text{cm}$ subtends an angle of $77^{\circ }$ at the centre of a circle. The radius of the circle is:

The relationship between the arc length ($s$), radius ($r$), and central angle ($\theta$ in radians) is given by:
$$s=r\theta$$

1. Convert the angle from degrees to radians:

$$\theta =77^{\circ }=77\times \frac{\pi }{180}\text{ radians}$$

2. Substitute the given values into the formula using $\pi \approx \frac{22}{7}$:

$$121=r\times \left(77\times \frac{22}{7\times 180}\right)$$

3. Simplify the expression inside the brackets:

$$121=r\times \left(11\times \frac{22}{180}\right)$$

$$121=r\times \frac{242}{180}$$

4. Solve for $r$:

$$r=\frac{121\times 180}{242}$$

$$r=\frac{1}{2}\times 180=90\text{cm}$$

Thus, the radius of the circle is $90\text{cm}$.