Solution

I : $x^2 - 23x + 120 = 0$

=> $x^2 - 8x - 15X + 120 = 0$

=> $x (x - 8) - 15 (x - 8) = 0$

=> $(x - 8) (x - 15) = 0$

=> $x = 8 , 15$

Thus, base = 8 CM and height = 15 cm (or vice versa)

=> Hypotenuse of right ANGLED triangle = $\sqrt{(8)^2 + (15)^2}$

= $\sqrt{64 + 225} = \sqrt{289} = 17 cm$

Since, triangle is inscribed in circle, => Radius of circle = half of hypotenuse

=> $r = \frac{17}{2} = 8.5$ cm

$\therefore$ Area of circle = $\PI r^2$

= $\frac{22}{7} \times 8.5 \times 8.5 \approx 227 cm^2$

Thus, I alone is sufficient.

Clearly, we cannot find base and height from statements II or III. Thus, they are insufficient.