A right-angled triangle is inscribed in a given circle. What is the area of the given circle (in cm2)? I. The base and height of the triangle (in cm) are both the roots of the equation x2 - 23x + 120 = 0 II. The sum of the base and height of the triangle is 23 cm. III. The height of the right-angled triangle is greater than the base of the same.

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Step-by-step solution:

1. Key fact: For a right-angled triangle inscribed in a circle, the hypotenuse is the diameter of the circle. Thus if the hypotenuse is $h$, the circle radius is $r=	frac{h}{2}$ and the area is $A=\pi r^2=\pi\left(	frac{h}{2}ight)^2=	frac{\pi h^2}{4}$.

2. Statement I: The base and height are the roots of $x^2-23x+120=0$. Factorising gives $(x-8)(x-15)=0$, so the legs are $8$ and $15$. The hypotenuse is $h=\sqrt{8^2+15^2}=17$. Hence the circle area is
$A=	frac{\pi\cdot 17^2}{4}=	frac{289\pi}{4}=72.25\pi\ 	ext{cm}^2$. Statement I therefore uniquely determines the area — it is sufficient.

3. Statement II: Only gives the sum of the legs $a+b=23$. Many positive pairs $(a,b)$ with sum $23$ are possible (for example $(15,8)$ and $(16,7)$), producing different hypotenuses and different circle areas. So II alone is insufficient.

4. Statement III: The inequality $	ext{height}>	ext{base}$ gives no magnitudes; alone it is insufficient.

5. Combining II and III still leaves multiple possible pairs (e.g. $(15,8)$ and $(16,7)$ both satisfy sum $23$ with one leg larger), so II+III is insufficient.

Therefore only statement I suffices to find the area.

FINAL_ANSWER: d

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