GMAT Math : Geometry

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Example Questions

Example Question #2 : Dsq: Calculating The Length Of The Side Of An Acute / Obtuse Triangle

Rect

Note: Figure NOT drawn to scale.

The above shows a triangle $\Delta MCT$ inscribed inside a rectangle $\textrm{Rect} \; RECT$ . is $\Delta MCT$ isosceles?

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

Statement 2: $\angle RTM \cong \angle ECM$

Possible Answers:

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.

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:

We show Statement 1 alone is sufficient:

If is the midpoint of $\overline{RE}$ , then MR = MC . Opposite sides of a rectangle are congruent, so RT = EC ; all angles of a rectangle, being right angles, are congruent, so $\angle MRT \cong \angle MEC$ . This sets up the conditions for the Side-Angle-Side Theorem, and $\Delta MRT \cong \Delta MEC$ . Consequently, MT = MC , and $\Delta MCT$ is isosceles.

Now, we show Statement 2 alone is sufficient:

If $\angle RTM$ , and $\angle ECM$ are congruent, then $\angle MTC$ and $\angle MCT$ , being complements of congruent angles, are congruent themselves. By the Isosceles Triangle Theorem, $\Delta MCT$ is isosceles.

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Example Question #3 : Dsq: Calculating The Length Of The Side Of An Acute / Obtuse Triangle

Which side of $\Delta ABC$ is the longest?

Statement 1: $\angle B$ is an obtuse angle.

Statement 2: $\angle A$ and $\angle C$ are both acute angles.

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.

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:

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

Explanation:

If we only know that two interior angles of a triangle are acute, we cannot deduce the measure of the third, or even if it is obtuse or right; therefore, Statement 2 alone does not help us.

If we know that $\angle B$ is an obtuse angle, however, we can deduce that $\angle A$ and $\angle C$ are both acute angles, since at least two interior angles of a triangle are acute. Therefore, we can deduce that $\angle B$ has the greatest measure, and that its opposite side, $\overline{AC}$ , is the longest.

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Example Question #4 : Dsq: Calculating The Length Of The Side Of An Acute / Obtuse Triangle

Is $\Delta ABC$ isosceles?

Statement 1: $\Delta ABC \sim \Delta DEF$

Statement 2: DE = EF

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 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 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 does not tell us anything unless we know the relative lengths of the sides of $\Delta DEF$ ; Statement 2 only gives us information about another triangle.

Suppose we assume both statements. Then by similarity,

$\frac{AB}{DE} = \frac{BC}{EF}$ .

Since DE = EF , then

$\frac{AB}{DE} \cdot DE = \frac{BC}{EF} \cdot EF$ , or

AB = BC .

This makes $\Delta ABC$ isosceles.

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Example Question #5 : Dsq: Calculating The Length Of The Side Of An Acute / Obtuse Triangle

Which of the three sides of $\Delta ABC$ is the longest?

Statement 1: $m \angle A =m \angle B + 20^{\circ }$

Statement 2: $m \angle C = m \angle A + 100^{\circ }$

Possible Answers:

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 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.

Correct answer:

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

Explanation:

The longest side of a triangle is opposite the angle of greatest measure.

From Statement 1 alone, we can find two possible scenarios with different answers:

Case 1:

$m \angle B = 20 ^{\circ }, m \angle A = 40 ^{\circ }, m \angle C = 120 ^{\circ }$

Case 2:

$m \angle C = 20 ^{\circ }, m \angle B = 70 ^{\circ }, m \angle A = 90 ^{\circ }$

In both cases, $m \angle A =m \angle B + 20^{\circ }$ , but in Case 1, $\overline{AB}$ is the longest side, and in Case 2, $\overline{BC }$ is the longest side.

From Statement 2 alone, however, we know that $m \angle C > 100^{\circ }$ , so $\angle C$ is obtuse and the other two angles are acute. That makes $\overline{AB}$ the longest side.

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Example Question #295 : Geometry

True or false: $\Delta ABC$ is scalene.

Statement 1: $\angle A \ncong \angle B$

Statement 2: $\overline{AB} \ncong \overline{BC}$

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.

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 insufficient to answer the question.

Explanation:

Assume both statements are true.

By definition, a scalene triangle has three noncongruent sides. Sides opposite noncongruent angles of a triangle are noncongruent, so as a consequence of Statement 1, $\overline{AC} \ncong \overline{BC}$ . Statement 2 alone establishes that $\overline{AB} \ncong \overline{BC}$ . However, the two statements together do not establish whether or not $\overline{AB} \ncong \overline{AC}$ , so it is not clear whether $\Delta ABC$ is scalene or isosceles.

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Example Question #5 : Dsq: Calculating The Length Of The Side Of An Acute / Obtuse Triangle

True or false: $\Delta ABC$ is scalene.

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

Statement 2: $\angle B \cong \angle C$

Possible Answers:

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.

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:

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

Explanation:

By definition, a scalene triangle has three noncongruent sides.

Statement 1 alone states that two sides are noncongruent, but no information is given about whether or not third side $\overline{AB}$ is congruent to either of the other sides.

Assume Statement 2 alone. In a triangle, sides opposite congruent angles are congruent, so it follows that $\overline{AB} \cong \overline{AC}$ . The triangle cannot be scalene.

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Example Question #292 : Geometry

True or false: $\Delta ABC$ is scalene.

Statement 1: AB > AC

Statement 2: BC > AC

Possible Answers:

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.

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 insufficient to answer the question.

Explanation:

By definition, a scalene triangle has three noncongruent sides.

If AB = 7, AC = 5, BC = 6 , then AB > AC and BC > AC , and the triangle is scalene.

If AB = 7, AC = 5, BC = 7 , then AB > AC and BC > AC , but AB = BC , so the triangle is not scalene.

The two statements together are insufficient.

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Example Question #291 : Geometry

Is $\Delta ABC$ an equilateral triangle?

Statement 1: $m\angle A =m\angle B = 60^{\circ }$

Statement 2: $\Delta ABC \sim \Delta DEF$ , and $\Delta DEF$ is equiangular.

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.

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.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

If $m\angle A =m \angle B = 60^{\circ }$ , then

$m\angle C = 180 - \left (m\angle A + m \angle B \right )=180 - \left (60 +60 \right ) = 60^{\circ }$ .

This makes $\Delta ABC$ an equiangular triangle.

If $\Delta ABC \sim \Delta DEF$ , and $\Delta DEF$ is equiangular, then, since corresponding angles of similar triangles are congruent, $\Delta ABC$ has the same angle measures, and is itself equiangular.

From either statement, since all equiangular triangles are equilateral, we can draw this conclusion about $\Delta ABC$ .

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Example Question #52 : Triangles

True or false: $\Delta ABC$ is equilateral.

Statement 1: The perimeter of $\Delta ABC$ is .

Statement 2: AB= 4 .

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.

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.

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 insufficient to answer the question.

Explanation:

The two statements together provide insufficient information. A triangle with sides , , and is equilateral and has perimeter ; A triangle with sides , , and is not equilateral and has perimeter .

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Example Question #53 : Triangles

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$\overline{DC}$ is the height of $\bigtriangleup ABC$ . What is the length of $\overline{AB}$ ?

(1) $\overline{DC}=4$

(2) $\overline{DB}=3$

Possible Answers:

Statements 1 and 2 together are not sufficient

Statement 2 alone is sufficient

Statement 1 alone is sufficient

Both statements together are sufficient

Each statement alone is sufficient

Correct answer:

Statements 1 and 2 together are not sufficient

Explanation:

To find the answer we should know more about the characteristics of the triangle, i.e. its angles, sides...

Statement 1 alone is obviously insufficient, since we don't know whether the triangle is equilateral, nothing can be said about AB.

Statement 2 is equally as unhelpful as statement 1, since we don't know whether ABC is of a specific type of triangle.

Taken together, these statements allow us to calculate the length of CB, but we can't go further, because we don't know what is AD.

Therefore statements 1 and 2 are not sufficient even taken together.

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GMAT Math : Geometry

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #2 : Dsq: Calculating The Length Of The Side Of An Acute / Obtuse Triangle

Example Question #3 : Dsq: Calculating The Length Of The Side Of An Acute / Obtuse Triangle

Example Question #4 : Dsq: Calculating The Length Of The Side Of An Acute / Obtuse Triangle

Example Question #5 : Dsq: Calculating The Length Of The Side Of An Acute / Obtuse Triangle

Example Question #295 : Geometry

Example Question #5 : Dsq: Calculating The Length Of The Side Of An Acute / Obtuse Triangle

Example Question #292 : Geometry

Example Question #291 : Geometry

Example Question #52 : Triangles

Example Question #53 : Triangles

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