# GMAT Math : Circles

## Example Questions

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### Example Question #5 : Dsq: Calculating The Ratio Of Diameter And Circumference

What is the circumference of the circle?

1. The diameter of the circle is .
2. The area of the circle is .
Possible Answers:

Each statement alone is sufficient to answer the question.

Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.

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

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

Statements 1 and 2 are not sufficient, and additional data is needed to answer the question.

Correct answer:

Each statement alone is sufficient to answer the question.

Explanation:

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.

### Example Question #1 : Dsq: Calculating The Length Of A Chord

Note: Figure NOT drawn to scale

Refer to the above figure. Is  an isosceles triangle?

Statement 1:  and  have equal length.

Statement 2:  and  have equal degree measure.

Possible Answers:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

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.

### Example Question #81 : Circles

Note: Figure NOT drawn to scale.

Give the length of chord .

Statement 1: Minor arc  has length .

Statement 2: Major arc  has length .

Possible Answers:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

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.

### Example Question #1 : Dsq: Calculating The Length Of A Chord

Note: Figure NOT drawn to scale.

In the above figure,  is the center of the circle, and  is equilateral. Give the length of Give the length of chord .

Statement 1: The circle has area .

Statement 2:  has perimeter .

Possible Answers:

BOTH statements TOGETHER are insufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

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.

### Example Question #4 : Chords

Note: Figure NOT drawn to scale.

Examine the above figure. True or false: .

Statement 1: Arc  is longer than arc .

Statement 2: Arc  is longer than arc .

Possible Answers:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

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.

### Example Question #5 : Chords

Note: Figure NOT drawn to scale

Examine the above figure. True or false: .

Statement 1:

Statement 2:

Possible Answers:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

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,

.

### Example Question #6 : Chords

Note: Figure NOT drawn to scale

Examine the above figure. True or false: .

Statement 1:

Statement 2:

Possible Answers:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

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.

### Example Question #7 : Chords

Note: Figure NOT drawn to scale.

Examine the above diagram. True or false: .

Statement 1:  is the midpoint of .

Statement 2:  is the midpoint of .

Possible Answers:

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

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 .

### Example Question #81 : Circles

Note: Figure NOT drawn to scale.

Examine the above figure. True or false: .

Statement 1: Arcs  and   have the same length.

Statement 2: Arcs  and   have the same degree measure.

Possible Answers:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

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.

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