In the figure, \(ABC\) is an equilateral triangle, and \(D\)

[BTGmoderatorLU]

Source: Manhattan Prep



In the figure, \(ABC\) is an equilateral triangle, and \(DBA\) is a right triangle. What is the area of the circumscribed circle?

1) \(DA=4\)
2) Angle \(ABD=30\) degrees.

The OA is A

[deloitte247]

$$\triangle DAB=90\ \deg ree$$
DB=diameter
$$\triangle ABD\ and\ \triangle ABC\ \ have\ same\ base\ AB$$
$$\triangle ABD\ and\ \triangle ABC\ \ have\ same\ base\ AB$$
AB = chord
$$\sin ce\ \triangle ACB=equilateral=60\deg$$
$$\triangle ADB=60\deg$$
Hence
$$\triangle DBA=1:2:\sqrt{3}$$

Statement 1
DA = 4
$$The\ side\ opposite\ 30\deg\ =\ 4$$
hypotenuse DB = diameter = 4*2 = 8
radius = 8/2 = 4
$$Area=\pi r^2=\pi4^2=16\pi=16\cdot\left(3.142\right)=50.37$$
Statement 1 is SUFFICIENT.

Statement 2
$$\triangle ABD=30\deg$$
There is no information about the sides of the triangle hence area of the inscribed circle cannot be evaluated
Statement 2 is NOT SUFFICIENT. Since statement 1 alone is SUFFICIENT.

$$Answer\ is\ Option\ A$$