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Inscribed Angles are equal

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by HeyArnold » Sun Jul 31, 2011 4:22 pm
https://postimage.org/image/31dw2muuc/

(sorry--could not get the image to show in this post! URL above)

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

[spoiler]OA: A the answer refers to the rule where two inscribed angles that cut the same arc are equal. The claim is they both cut the arc ACB... this is nearly the entire circle, how can you tell where the arcs "end" for this rule?

Thanks![/spoiler]
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Source: — Data Sufficiency |
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by edge » Sun Jul 31, 2011 5:09 pm
St1 and St2 are insufficient by themselves.

It is a property of any circle that its diameter inscribes a right triangle at all points on the curve. Therefore, DB is the diameter. From the two statements, we know that AD = 4 and ∠B = 30°. Therefore, DB = 8 (using basic trigonometry) and the area of the circle is 16Π.

The answer is C.
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by bblast » Mon Aug 01, 2011 6:13 am
Hi,
Remember 2 rules :

1>angle subtended by an arc at all points of a circle are always equal. Thus in the question at hand. Minor arc AB subtends angle C = angle D = 60 degree. Thus statement 1 provides us with sufficient information. Statement 2 simply reiterates what we have from the problem statement.

2> Additionally remember that the angle subtended by arc AB at the center of the circle will be twice that subtended at the circumference(here that would be equal to 120 degree). This is also a frequently tested concept.
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by GMATGuruNY » Mon Aug 01, 2011 3:54 pm
HeyArnold wrote:https://postimage.org/image/31dw2muuc/

(sorry--could not get the image to show in this post! URL above)

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

[spoiler]OA: A the answer refers to the rule where two inscribed angles that cut the same arc are equal. The claim is they both cut the arc ACB... this is nearly the entire circle, how can you tell where the arcs "end" for this rule?

Thanks![/spoiler]
Image

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:

Image

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

The correct answer is A.
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