Chords - GMAT Quantitative

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 .

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 .

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.