gmattesttaker2 wrote:Hello,

Can you please assist with the following:

In the figure, ABC is an equilateral triangle, and DAB is a right triangle. What is the area of the circumscribed circle?

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

OA: A

I was just wondering in statement 1, how we can find out if ADB is a 30 60 90 triangle?

Thanks for your help.

Regards,
Sri

Since ∆ABC is equilateral, ∠ACB = 60º.

An inscribed angle is formed by TWO CHORDS.
Inscribed angles that intercept the same two points on a circle are EQUAL.
Inscribed angle ADB (in red) and inscribed angle ACB (in blue) both intercept the circle at points A and B.
Thus, ∠ADB = ∠ACB = 60º.

Since ∆ABC is equilateral, the result is the following figure:


As the figure shows, ∆ABD is a 30-60-90 triangle.
In a 30-60-90 triangle, the sides are in the following ratio:
x : x√3 : 2x.
Since AD : AB : BD = x : x√3 : 2x, diameter BD = 2(AD).

Statement 1: AD = 4
Thus, diameter BD = 2*4 = 8.
Since the diameter of the circle is known, the area of the circle can be determined.
SUFFICIENT.

Statement 2: Angle ABD = 30 degrees
No new information is offered by this statement.
The second figure above already indicates that ∠ABD = 30º.
INSUFFICIENT.

The correct answer is A.