M20-07

Official Solution:


(1) \(AB^2 = BC^2 + AC^2\). This means that triangle \(ABC\) is a right triangle with \(AB\) as hypotenuse, so \(area=\frac{BC*AC}{2}\). Now, a right triangle inscribed in a circle must have its hypotenuse as the diameter of the circle (the reverse is also true: if the diameter of the circle is also the triangle’s side, then that triangle is a right triangle). So, hypotenuse \(AB=diameter=2*radius=2\), but just knowing the length of the hypotenuse is not enough to calculate the legs of a right triangle thus we can not get the area. Not sufficient.

(2) \(\angle CAB\) equals 30 degrees. Clearly insufficient.

(1)+(2) From (1) \(ABC\) is a right triangle and from (2) \(\angle CAB=30\). Hence we have 30°-60°-90° right triangle and as \(AB=hypotenuse=2\) then the legs equal to 1 and \(\sqrt{3}\) (in 30°-60°-90° right triangle the sides are always in the ratio \(1:\sqrt{3}:2\)). Therefore \(area=\frac{BC*AC}{2}=\frac{\sqrt{3}}{2}\). Sufficient.


Answer: C