Data Sufficiency

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

rommysingh wrote: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.

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

Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and equations ensures a solution.


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.


If you know our own innovative logics to find the answer, you don't need to actually solve the problem.
www.mathrevolution.com
l The one-and-only World's First Variable Approach for DS and IVY Approach for PS that allow anyone to easily solve GMAT math questions.

l The easy-to-use solutions. Math skills are totally irrelevant. Forget conventional ways of solving math questions.

l The most effective time management for GMAT math to date allowing you to solve 37 questions with 10 minutes to spare

l Hitting a score of 45 is very easy and points and 49-51 is also doable.

l Unlimited Access to over 120 free video lessons at https://www.mathrevolution.com/gmat/lesson

l Our advertising video at https://www.youtube.com/watch?v=R_Fki3_2vO8