Points A and B lie on Circle O. Is the area of Circle O greater than 8

Answer is B, since statement (1) is not sufficient as we require the arc angle in order to determine the radius and hence area.

I believe there is a way to figure this out using geometry that is too advanced for me

1)Length of arc = some formula allowing us to back solve but I don't think it would be enough alone

2) length of chord AB is 18.. Ok, I feel like applying that to the formula I don't know but mentioned above could work

Note to self: I should probably google some formulas about chords and arcs now

statement 1 -
a formula is there for this but it has 2 unknowns radious and angle at centre both not available . so not enough

Statement 2 -
also not enough alone

I dont remember any method to find even taking both statements together
So marking E

Points A and B lie on Circle O

Is the area of Circle O > 80π
Is \(\pi r^2 > 80\pi\)
Is \(r^2 > 80\)

Statement (1):
The length of minor arc AB is \(6\pi\).

if minor arc's length is a fraction less than half the circumference of circle O,
then major arc's length is a fraction more than half the circumference
circumference \(\approx\) minor arc + major arc \(\approx\) \(6\pi\) + \(6\pi\)
\(2\pi r\) = \(12\pi\)
\(r^2\) = 36 < 80

if minor arc's length is one-fourth length of the circumference of circle O,
then major arc's length is three-fourth length of the circumference
circumference = minor arc + major arc = \(6\pi\) + \(18\pi\)
\(2\pi r\) = \(24\pi\)
\(r^2\) = 144 > 80

Not Sufficient

Statement (2):
The length of chord AB is 18

Diameter is the largest chord of a circle

If AB is diameter of a circle, then it will give the MINIMUM area of the circle O
radius \(r\) = \(\frac{AB}{2}\)
r = \(\frac{18}{2}\)
\(r^2\) = 81 > 80

Sufficient

Answer B

Hi All,

This DS question requires a little bit of Geometry "math" and the knowledge of a couple of very specific Geometry terms.

We're told that points A and B lie on a Circle. We're asked if the AREA of the circle is GREATER than 80pi. This is a YES/NO question.

Fact 1: The length of MINOR ARC AB is 6pi.

By definition, a 'minor arc' is LESS than 180 degrees of the circle.

IF....
the minor arc 6pi was 179.9999 degrees of the circle...
then the circumference of the circle would be almost 12pi
and the radius would be almost 6
the area would be almost 36pi and the answer to the question is NO.

IF....
the minor arc 6pi was 30 degrees of the circle...
then the circumference of the circle would be 72pi
and the radius would be 36
the area would be almost (36^2)pi and the answer to the question is YES.
Fact 1 is INSUFFICIENT

Fact 2: The length of chord AB is 18.

By definition, a "chord" is a straight line that connects any two points on the circumference of a circle. The LARGEST chord is the diameter.

IF....
the chord 18 IS the diameter....
then the radius is 9
the area is 81pi and the answer to the question is YES

IF....
the chord 18 is NOT the diameter....
then the diameter is GREATER than 18
and then the radius is GREATER than 9
the area is GREATER than 81pi and the answer to the question is STILL YES

With the information in Fact 2, the answer to the question is ALWAYS YES.
Fact 2 is SUFFICIENT

Final Answer:

GMAT assassins aren't born, they're made,
Rich

VERITAS PREP OFFICIAL SOLUTION:

Correct Answer: (B)

From Statement (1), given that 6π is the length of the minor arc, we can assume that the major arc is larger, but this still only guarantees a circumference that is greater than 12π. This means that the radius must be greater than 6, which in turn guarantees an area greater than 36π. The area may still be either less than or greater than 80π, so Statement (1) is insufficient. Now look at Statement (2). If AB is the diameter of the circle, the radius is 9, and the area is 81π. If AB is a chord but not the diameter, the diameter will be larger than 18, and the area will be larger than 81π. In any case, if AB = 18, the area of the circle will be greater than 80π. So Statement (2) alone is sufficient.

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