Chords - GMAT Quantitative Reasoning

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Question

Note: Figure NOT drawn to scale

Refer to the above figure. Is $\Delta ABC$ an isosceles triangle?

Statement 1: $\widehat{ABC}$ and $\widehat{BCA}$ have equal length.

Statement 2: $\widehat{ABC}$ and $\widehat{BCA}$ have equal degree measure.

Answer

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 $\widehat{ABC}$ and $\widehat{BCA}$ have equal degree measure, then their minor arcs $\widehat{AC}$ and $\widehat{BA}$ do also. Congruent arcs on the same circle have congruent chords, so $\overline{AC} \cong \overline{ BA}$ , and this proves $\Delta ABC$ isosceles.

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Question

Note: Figure NOT drawn to scale.

Give the length of chord $\overline{AB}$ .

Statement 1: Minor arc $\widehat{AB}$ has length $8 \pi$ .

Statement 2: Major arc $\widehat{ACB}$ has length $16 \pi$ .

Answer

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 $8 \pi + 16 \pi = 24 \pi$ . It follows that the radius is $24 \pi \div 2 \pi = 12$ , and that $n =\frac{ 8 \pi }{24 \pi } \times 360 ^{\circ } = 120 ^{\circ }$ . 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.

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Question

Note: Figure NOT drawn to scale.

In the above figure, is the center of the circle, and $\Delta OAB$ is equilateral. Give the length of Give the length of chord $\overline{AB}$ .

Statement 1: The circle has area $64 \pi$ .

Statement 2: $\Delta OAB$ has perimeter .

Answer

Since $\Delta OAB$ is equilateral, the length of chord $\overline{AB}$ is equivalent to the length of $\overline{OA}$ , 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 $\overline{AB}$ is one third of the known perimeter.

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Question

Note: Figure NOT drawn to scale.

Examine the above figure. True or false: $\overline{AB} \cong \overline{CD}$ .

Statement 1: Arc $\widehat{CAB}$ is longer than arc $\widehat{ACD}$ .

Statement 2: Arc $\widehat{ABD}$ is longer than arc $\widehat{CDB}$ .

Answer

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 $\widehat{CAB}$ is the sum of the lengths of $\widehat{CA }$ and $\widehat{ AB}$ , which we will call and , respectively. Similarly, the length of $\widehat{ACD}$ is the sum of the lengths of $\widehat{AC}$ and $\widehat{ CD}$ , which we will call and , respectively.

If Statement 1 alone is assumed,

Subsequently,

,

so $\widehat{ AB}$ is longer than $\widehat{ CD}$ . 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.

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Question

Note: Figure NOT drawn to scale.

Examine the above figure. True or false: $\overline{AB} \cong \overline{CD}$ .

Statement 1: Arcs $\widehat{ AB}$ and $\widehat{ CD}$ have the same length.

Statement 2: Arcs $\widehat{ADB}$ and $\widehat{CBD}$ have the same degree measure.

Answer

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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Question

Note: Figure NOT drawn to scale

Examine the above figure. True or false: $\overline{AB} \cong \overline{YZ}$ .

Statement 1: $\overline{XY} \cong \overline{BC}$

Statement 2: $\angle ABC \cong \angle XYZ$

Answer

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 $m \widehat{AC} = m \widehat{ZX}$ - 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 $\overline{XY} \cong \overline{BC}$ , then $m \widehat{CB} = m \widehat{XY}$ . From Statement 2, as stated before, $m \widehat{AC} = m \widehat{ZX}$ . Then,

$m \widehat{AC} + m \widehat{CB} = m \widehat{ZX} + m \widehat{XY}$

By arc addition, this statement becomes

$m \widehat{AB} = m \widehat{ZY}$ .

Since congruent chords on the same circle have congruent arcs,

$\overline{AB} \cong \overline{YZ}$ .

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Question

Note: Figure NOT drawn to scale

Examine the above figure. True or false: $\overline{AB} \cong \overline{YZ}$ .

Statement 1: $\overline{XY} cong \overline{BC}$

Statement 2: $\angle ABC cong \angle XYZ$

Answer

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

$\overline{AB} \cong \overline{YZ}$ if and only if $m \widehat{AB} = m \widehat{ZY}$ .

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

$m \widehat{BC} e m \widehat{XY}$ .

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

$m \widehat{AC} e m \widehat{XZ}$

By arc addition,

$m \widehat{AB} = m \widehat{ZY}$

can be expressed as

$m \widehat{AC} +m \widehat{BC} = m \widehat{XZ} + m \widehat{XY}$ .

Examples of the values of the four arc measures $m \widehat{AC}$ , $m \widehat{XZ}$ , $m \widehat{BC}$ , and $m \widehat{XY}$ can easily be found to make $m \widehat{AC} e m \widehat{XZ}$ and $m \widehat{BC} e m \widehat{XY}$ so that

$m \widehat{AC} +m \widehat{BC} = m \widehat{XZ} + m \widehat{XY}$ is either true or false; consequently, $\overline{AB} \cong \overline{YZ}$ may be true or false.

The two statements together are insufficient.

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Question

Note: Figure NOT drawn to scale.

Examine the above diagram. True or false: $\overline{AB} \cong \overline{CD}$ .

Statement 1: is the midpoint of $\overline{BD}$ .

Statement 2: is the midpoint of $\overline{AC}$ .

Answer

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,

$\Delta ABM \sim \Delta DCM$

$\overline{AB} \cong \overline{CD}$ 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 $\overline{BM} \cong \overline{DM}$ , and from Statement 2 alone, it follows that $\overline{AM} \cong \overline{CM}$ . 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 $\Delta ABM \cong \Delta DCM$ , and, subsequently, that $\overline{AB} \cong \overline{CD}$ .

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