Bunuel wrote:
In the diagram above, O is the center of the circle and angle AOB = 144º. What is the area of the circle?
(1) The area of sector AOB is 40% of the area of the circle
(2) Arc ACB has a length of \(14\pi\).
MAGOOSH OFFICIAL SOLUTION:First of all, we should think about the prompt a bit. This angle, 144º, is what fraction of a full circle? Well, both 144 and 360 are divisible by 4: 144 ÷ 4 = 36 and 360 ÷ 4 = 90, so 144/360 = 2/5.
This angle is 2/5 of the whole circle, so the arc is 2/5 of the whole circumference. Also, remember that if we can find anything about the whole circle, for example, the circumference, then we can find the radius, which would allow us to find the area.
Statement #1: This is a tautological statement. A tautological statement is a statement that, by definition, has to be true, and because of this, it contains no information. Statements such as “My car is a car” and “My employer employs me” are verbal tautologies: they contain no useful information. Much in the same way, we already know from the prompt that the angle takes up 2/5 of the circle, so of course the sector would take up 2/5, or 40%, of the area. This statement repeats information in the prompt, and contains no new information, so it doesn’t help us at all to figure out anything else. This statement, alone and by itself, is not sufficient.
Statement #2: We already know this arc is 2/5 of the whole circumference, so we could set up a proportion to find the circumference. From that, we could find the radius, and that would allow us to find the area. This statement, alone and by itself, is sufficient.
Answer = (B)
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