Circles - GMAT Quantitative

Statement 1: If we know the circumference, we can calculate the diameter.

If then

Statement 2: We know the diameter of circle B is and that the ratio of the circles. We can set up our proportions and find the diameter:

Statement 1: We're given a ratio (which you should already know) but no values. We need additional information to answer the question.

Statement 2: If we're given the circumference, we can solve for the diameter.

which means

Statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question.

Statement 1: We can calculate the circumference using the given diameter.

Statement 2: To find the circumference, we must first find the radius of the circle using the given area.

We can plug this value into the equation for circumference:

Each statement alone is sufficient to answer the question.

We are asked to find the ratio of the circumference to the diameter of Circle and are given the radius and the area. We also know that . Taking each statement individually:

1.) The radius is and we know that . The diameter . Since we can determine that , Statement 1 is sufficient to solve for the ratio by itself.

2.) The area of the outside circle is , so therefore . Since we can use to determine both the circumference and diameter , Statement 2 is sufficient to solve for the ratio by itself.

Statement 1 and Statement 2 are equivalent, as two arcs on the same circle have the same length if and only if they have the same degree measure. We only need to prove the sufficiency or insufficiency of one statement to answer the question.

Choose Statement 2. If and have equal degree measure, then their minor arcs and do also. Congruent arcs on the same circle have congruent chords, so , and this proves isosceles.

Statement 1 alone is insufficient to give the length of the chord, since no other information is known about the major arc, the circle, or the angle. For similar reasons, Statement 2 alone is insufficient.

If both statements are assumed, then it is possible to add the arc lengths to get the circumference of the circle, which is . It follows that the radius is , and that . From this information, can be calculated by bisecting the triangle into two 30-60-90 triangles with a perpendicular bisector from , and applying the 30-60-90 theorem.

Since is equilateral, the length of chord is equivalent to the length of , and, subsequently, the radius of the circle. If Statement 1 alone is assumed, the radius of the circle can be calculated using the area formula.

If Statement 2 alone is assumed, the length of is one third of the known perimeter.

For two chords in the same circle to be congruent, it is necessary and sufficient that their arcs have the same length.

By arc addition, the length of is the sum of the lengths of and , which we will call and , respectively. Similarly, the length of is the sum of the lengths of and , which we will call and , respectively.

If Statement 1 alone is assumed,

Subsequently,

,

so is longer than . The arcs are of unequal length so their chords are as well. This makes Statement 1 sufficient to answer the question. A similar argument can be made that Statement 2 alone answers the question.

For two chords in the same circle to be congruent, it is necessary and sufficient that their minor arcs have the same length. Statement 1 asserts this, so it is sufficient to answer the question.

It is also necessary and sufficient that their major arcs have the same degree measure. Statement 2 alone asserts this, so it is sufficient to answer the question.

Statement 1 only gives information about two other chords, whose relationship with the first two is not known. Statement 2 only gives the congruence of two inscribed angles - and, subsequently, since congruent inscribed angles intercept congruent arcs, that - but gives no information about the individual sides.

Assume both statements.

Congruent chords of the same circle must have arcs of the same degree measure, so, from Statement 1, since , then . From Statement 2, as stated before, . Then,

By arc addition, this statement becomes

.

Since congruent chords on the same circle have congruent arcs,

.

Congruent chords of the same circle must have arcs of the same degree measure, so

if and only if .

Assume both statements. Then , since, in the same circle, congruent arcs have congruent chords, it follows from Statement 1 that

.

Also, since congruent inscribed angles intercept congruent arcs, it follows from Statement 2 that

By arc addition,

can be expressed as

.

Examples of the values of the four arc measures , , , and can easily be found to make and so that

is either true or false; consequently, may be true or false.

The two statements together are insufficient.

If two chords of a circle intersect inside it, and two more chords are constructed connecting endpoints, as is the case here, the resulting triangles are similar - that is,

if and only if the triangles are congruent. From either statement alone, we are given a side congruence - from Statement 1 alone it follows that , and from Statement 2 alone, it follows that . Either way, the resulting side congruency, along with two angle congruencies following from the similarity of the triangles, prove by way of the Angle-Sude-Angle Postulate that , and, subsequently, that .

Let be the radius of Circle B. Then Circle A has radius and, subsequently, area . Since the area of Circle B is , the area of Circle A is 36 times that of Circle B.

The given sector of Circle A has the same area as Circle B, so the sector is one thirty-sixth of the circle. That makes the angle measure of the sector

The radius of Circle A and the length of a side of the square are the same - we will call each . The area of the circle is ; that of the square is . Therefore, a sector of the circle with area will be of the circle, which is a sector of measure

The area of a sector of radius is

From the first statement alone, you can halve the diameter to get radius 24 inches.

From the second alone, note that the length of the arc is

Given that length, you can find the radius:

Either way, you can get the radius, so you can calculate the area.

The answer is that either statement alone is sufficient to answer the question.

The width of the rectangle is equal to the radius of the quarter circles, which we call ; the length is twice that, or .

The area of the rectangle is ; the total area of the two black quarter circles is , so the area of the white region is their difference,

Therefore, all that is needed to find the area of the white region is the radius of the quarter circle.

If we know that the area of the black region is centimeters, then we can deduce using this equation:

If we know that the perimeter of the rectangle is 60 centimeters, we can deduce via the perimeter formula:

Either statement alone allows us to find the radius and, consequently, the area of the white region.

The time alone is insufficient without the length of the hand. The second statement does not give us that information, only the relationship between the lengths of the two hands, which is useless without the length of the minute hand.

The answer is that both statements together are insufficient to answer the question.

Since there are twelve numbers on the clock, the angular measure from one number to the next is ; this means represents two and a half number positions.

Suppose we know both statements. Since the minute hand is on the 6, the hour hand is either midway between the 3 and the 4, or midway between the 8 and the 9. Both scenarios are possible, as they correspond to 3:30 and 8:30, respectively, so the question is not answered even if we know both statements.

The first event happens numerous times over the course of twelve hours, so the first statement is not enough to deduce the time; all the second statement tells you is that it is forty minutes after an hour (12:40, 1:40, etc.)

Suppose we put the two statements together. It is from one number to the next, so the 8:00 position is the position. If the hour hand makes a with the minute hand, then the hour hand is either at or . Since forty minutes is two-thirds of an hour, however, the hour hand must be two-thirds of the way from one number to the next.

Case 1: If the hour hand is at , then it is at the position - in other words, two-thirds of the way from the 5 to the 6. This is consistent with our conditions.

Case 2: If the hour hand is at , then it is at the position - in other words, one-third of the way from the 10 to the 11. This is inconsistent with our conditions.

Therefore, only the first case is possible, and if we are given both statements, we know it is 5:40.

The two statements together are not enough unless you know the size of the minute hand; without this information, you cannot tell the angular position of the minute hand, so, even if you know the angle the hands are making, you do not know the position of the hour hand either.