GMAT Math : Geometry

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Example Questions

Example Question #4 : Diameter

Circle A and circle B are given. If the diameter of circle B is , what is the diameter of circle A?

The circumference of circle A is $52\pi$ .
The ratio of the diameters of circle A and circle B is , respectively.

Possible Answers:

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

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

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

Each statement alone is sufficient to answer the question.

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

Correct answer:

Each statement alone is sufficient to answer the question.

Explanation:

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

$C=2\pi r= d\pi$

If $C=52\pi$ then d=52

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:

$\frac{x}{13} =\frac{4}{1}$

x=4(13)=52

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Example Question #4 : Dsq: Calculating The Ratio Of Diameter And Circumference

What is the diameter of the circle?

The diameter to radius ratio is .
The circumference is $10 \pi$ .

Possible Answers:

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

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

Each statement alone is sufficient to answer the question.

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.

Correct answer:

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

Explanation:

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.

$C=2r\pi =d\pi$

$C=10\pi =d\pi$

which means d=10

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

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

What is the circumference of the circle?

The diameter of the circle is .
The area of the circle is $64\pi$ .

Possible Answers:

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

Each statement alone is sufficient to answer the question.

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

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

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.

$circumference=d\pi =16\pi$

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

$64\pi =r^2\pi$

r^2=64

r=8

We can plug this value into the equation for circumference:

$C=2r\pi=16\pi$

Each statement alone is sufficient to answer the question.

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Example Question #1 : Chords

Chords

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.

Possible Answers:

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.

BOTH statements TOGETHER are insufficient 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:

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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Example Question #511 : Geometry

Chord

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 $16 \pi$ .

Possible Answers:

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.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 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 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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Example Question #512 : Geometry

Chord

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 .

Possible Answers:

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.

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

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

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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Example Question #4 : Dsq: Calculating The Length Of A Chord

Chord

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 $\widehat{CDB}$ .

Possible Answers:

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.

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.

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 $\widehat{CAB}$ 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,

x+y > x+ z

Subsequently,

y > z ,

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

Chord

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: $\angle ABC \cong \angle XYZ$

Possible Answers:

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.

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 1 ALONE is sufficient to answer the question, but Statement 2 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 $m \widehat{AC} = m \widehat{ZX}$ - 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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Example Question #515 : Geometry

Chord

Note: Figure NOT drawn to scale

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

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

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

Possible Answers:

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.

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.

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

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

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

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

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

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 $m \widehat{AC} \ne m \widehat{XZ}$ and $m \widehat{BC} \ne m \widehat{XY}$ 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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Example Question #2 : Dsq: Calculating The Length Of A Chord

Chord

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 $\overline{AC}$ .

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.

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.

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,

$\Delta ABM \sim \Delta DCM$

$\overline{AB} \cong \overline{CD}$ 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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GMAT Math : Geometry

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #4 : Diameter

Example Question #4 : Dsq: Calculating The Ratio Of Diameter And Circumference

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

Example Question #1 : Chords

Example Question #511 : Geometry

Example Question #512 : Geometry

Example Question #4 : Dsq: Calculating The Length Of A Chord

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

Example Question #515 : Geometry

Example Question #2 : Dsq: Calculating The Length Of A Chord

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