Inscribed circle...MGMAT

[rommysingh]

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

[GMATGuruNY]

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

Inscribed ∠DAB = 90 degrees.
An inscribed angle of 90 degrees intercepts the diameter.
Thus, DB is the diameter of the circle.

Inscribed angles that intercept the same arc are equal.
Inscribed angles ∠ADB and ∠ACB both intercept arc AB.
Thus, ∠ADB = ∠ACB.
Since ∆ABC is equilateral, ∠ACB = 60 degrees, implying that ∠ADB = 60 degrees.
The result, as shown above, is that both ∆ADE and ∆AEB are 30-60-90 triangles.
Since the two triangles share side AE, if we know one side of either triangle, we can determine the lengths of all the other sides -- including DE and EB, which form the diameter.
The length of diameter DB will allow us to determine the area of the circle.

Question rephrased: What is the length of one side of either ∆ADE or ∆AEB?

Statement 1: AD = 4.
Sufficient.
See below:



Statement 2: ∠ABD = 30.
No new information.
The question stem itself implies that ∠ABD = 30.
Insufficient.

The correct answer is A.

[rommysingh]

Thanks for the reply,but i couldn't understand the idea of intercept the same arc...like how are the angles intercepting the same arc. I might be sounding dumb but then I am not able to get that...plz help

thanks

[Max@Math Revolution]

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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

In the original condition, the angle of circumference is identical and Angle ADB = 60 degrees.(Since triangle ABC is equilateral and angle C is 60 degrees , angle C = angle ADB). And since the central angle is double the angle of circumference, DB is the diameter. Then, since with particular angles DB = 2DA (1:root3:2), we only need to find out DA and since we have 1 variable, D is likely the answer. (We get the answer once we match the number of equations and number of variables. condition (1) & (2) have 1 each, thus D is likely the answer.)
In case of (1), DB=8 makes it sufficient, but (1) is just a replicate of the original condition, thus it is not sufficient. Therefore the answer is A.


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