GMAT Math : Geometry

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Example Questions

Example Question #2481 : Gmat Quantitative Reasoning

What is the perimeter of $\bigtriangleup ABC$ ?

Statement 1: The triangle with its vertices at the midpoints of $\overline{AB}$ , $\overline{AC}$ , and $\overline{BC}$ has perimeter 34.

Statement 2: , , and are the midpoints of the sides of a triangle with perimeter 136.

Possible Answers:

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.

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:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

Assume Statement 1 alone. A segment that has as its endpoints the midpoints of two sides of a triangle is a midsegment of the triangle, and its length is half that of the side to which it is parallel. Therefore, the sum of the lengths of the midsegments—that is, the perimeter of the triangle they form, which from Statement 1 is 34, is half the perimeter of the larger triangle, which here is $\bigtriangleup ABC$ . The perimeter of $\bigtriangleup ABC$ is therefore 68.

Assume Statement 2 alone. Here, $\bigtriangleup ABC$ itself is the triangle formed by the midsegments. Since the larger triangle has perimeter 136, $\bigtriangleup ABC$ has perimeter half this, or 68.

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Example Question #12 : Dsq: Calculating The Perimeter Of An Acute / Obtuse Triangle

True or false: The perimeter of $\bigtriangleup ABC$ is greater than 60.

Statement 1: $\bigtriangleup ABC$ is an isosceles triangle.

Statement 2: AB = 16 and BC = 24

Possible Answers:

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.

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.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

Assume both statements to be true. An isosceles triangle has two sides of equal length, so either AC= AB = 16 or AC= BC = 24 . By the Triangle Inequality Theorem, the length of each side must be less than the sum of the lengths of the other two; both scenarios are possible, since

16+16 = 32 > 24

and

16+24= 40> 24 .

If AC = 16 , the perimeter of $\bigtriangleup ABC$ is

p =AB + BC + AC

p =16+24+16 = 56 .

If AC = 24 , the perimeter of $\bigtriangleup ABC$ is

p =AB + BC + AC

p =16+24+24 = 64 .

Without further information, it is impossible to determine whether the perimeter is less than or greater than 60.

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Example Question #12 : Acute / Obtuse Triangles

What is the perimeter of $\bigtriangleup ABC$ ?

Statement 1: AB = 10 and BC = 10

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

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 sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

From Statement 1 alone, we know that two sides are of length 10, but we are not given any clue as to the length of the third. Therefore, we cannot calculate the sum of the side lengths, which is the perimeter. Statement 2 alone is unhelpful, since it only gives that two angles have equal measure; applying the Converse of the Isosceles Triangle Theorem, it can be determined that their opposite sides $\overline{AC}$ and $\overline{AB}$ are congruent, but no actual side lengths can be found.

Now assume both statements to be true. From Statement 1, $\overline{AB} \cong \overline{BC}$ , and, as stated before, it follows from Statement 2 that $\overline{AB} \cong \overline{AC}$ . Therefore, the triangle is equilateral with sides of common length 10, making the perimeter 30.

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Example Question #1 : Dsq: Calculating Whether Acute / Obtuse Triangles Are Congruent

AB = DE and AC = DF

Is it true that $\Delta ABC \cong \Delta DEF$ ?

1) $\angle B \cong \angle E$

2) $\angle C \cong \angle F$

Possible Answers:

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

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.

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:

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

Explanation:

If only one of the statements is known to be true, the only congruent pairs that are known between the triangles comprise two sides and a non-included angle; this information cannot prove congruence between the triangles. If both are known to be true, however, they, along with either of the given side congruences, set up the conditions for the Angle-Angle-Side Theorem, and the triangles can be proved congruent.

The answer is that both statements together are sufficient to answer the question, but not either alone.

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Example Question #1 : Dsq: Calculating Whether Acute / Obtuse Triangles Are Congruent

You are given two triangles $\Delta ABC$ and $\Delta DEF$ ; with $\overline{AB} \cong \overline{DE}$ and $\overline{BC} \cong \overline{EF}$ . Which side is longer, $\overline{AC}$ or $\overline{DF}$ ?

Statement 1: $m \angle B > m \angle E$

Statement 2: $\angle B$ and $\angle F$ are both right angles.

Possible Answers:

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

EITHER 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:

We are given two triangles with two side congruences between them. If we compare their included angles (the angles that they form), the angle that is of greater measure will have the longer side opposite it. This is known as the Hinge Theorem.

The first statement says explicitly that the first included angle, $\angle B$ , has greater measure than the second, $\angle E$ , so the side opposite $\angle B$ , $\overline{AC}$ , has greater measure than $\overline{DF}$ .

The second statement is not so explicit. But if $\angle F$ is a right angle, $\angle E$ must be acute, and if $\angle B$ is right, then $m \angle B > m \angle E$ , which again proves that AC > DF .

The answer is that either statement alone is sufficient to answer the question.

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Example Question #21 : Triangles

In $\bigtriangleup ABC$ , if $AB=k$ , $BC=k+3$ and $AC=m$ , which of the three angles of $\triangle ABC$ has the greatest degree measure?

(1) $k=3$

(2) $m=k+4$

Possible Answers:

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Correct answer:

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

Explanation:

The longest side is opposite the largest angle for all triangles.

(1) Substituting 3 for $k$ means that $AB=3$ and $BC=5$ . But the value of $m$ given for side $AC$ is still unknown $\rightarrow$ NOT sufficient.

(2) Since $k+3>k$ , the longest side must be either $k+3$ or $m$ . So, knowing whether $m>k+3$ is sufficient.

If $m=k+4$ , knowing that $k+4>k+3$ ,

then $m>k+3$ $\rightarrow$ SUFFICIENT.

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Example Question #22 : Triangles

A triangle contains a $110^{\circ}$ angle. What are the other angles in the triangle?

(1) The triangle is isosceles.

(2)The triangle has a perimeter of 12.

Possible Answers:

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

Statement (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

EACH statement ALONE is sufficient.

Correct answer:

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Explanation:

Statement 1: An isosceles triangle has two equal angles. Since the interior angles of a triangle always sum to $180^{\circ}}$ , the only possible angles the other sides could have are $35^{\circ}$ .

Statement 2: This does not provide any information relevant to the question.

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Example Question #3 : Dsq: Calculating An Angle In An Acute / Obtuse Triangle

Triangle_b

Note: figure NOT drawn to scale.

$m \angle A = 46 ^{\circ }$ . What is $m \angle 1$ ?

Statement 1: $\overline{AB} \cong \overline{AC}$

Statement 2: $m \angle B = 67 ^{\circ }$

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 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.

BOTH statements TOGETHER are insufficient to answer the question.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

If $\overline{AB} \cong \overline{AC}$ , then by the Isosceles Triangle Theorem, $m \angle B = m \angle 2$ . Since the sum of the measures of a triangle is 180,

$m \angle A + m \angle B + m \angle 2 = 180$

After some substitution,

$46 + m \angle 2 + m \angle 2 = 180$

$46 + 2m \angle 2 = 180$

$2m \angle 2 = 134$

$m \angle 2 = 67$

Since $\angle 1$ and $\angle 2$ form a linear pair,

$m \angle 1 + m \angle 2 = 180$ , and

$m \angle 1 = 180 - m \angle 2 = 180 - 67 = 113^{\circ }$

If $m \angle B = 67 ^{\circ }$ , then by the Triangle Exterior Angle Theorem,

$m \angle 2 = m \angle A + m \angle B = 46 + 67 = 113^{\circ }$

So either statement by itself provides sufficient information.

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Example Question #4 : Dsq: Calculating An Angle In An Acute / Obtuse Triangle

In $\triangle PQR$ above, what is the value of ?

(1) x=1.5y

(2) x+y=112.5

Possible Answers:

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

EACH statement ALONE is sufficient.

Correct answer:

EACH statement ALONE is sufficient.

Explanation:

There is an implied condition: 2x+y=180 . Therefore, with each statement, we have 2 unknown numbers and 2 equations. In this case, we can take a guess that we will be able to find the value of by using each statement alone. It’s better to check by actually solving this problem.

For statement (1), we can plug 1.5y into 2x+y=180 . Now we have 4y=180 , which means y=45 .

For statement (2), we can rewrite the equation to be x=112.5-y and then plug into 2x+y=180 , making it

$2\cdot \left ( 112.5-y \right )+y=180$

Then we can solve for and get y=45 .

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Example Question #5 : Dsq: Calculating An Angle In An Acute / Obtuse Triangle

Is triangle $\Delta ABC$ acute, right, or obtuse?

Statement 1: $m\angle A + m\angle B = 120^{\circ }$

Statement 2: $m\angle B + m\angle C = 110^{\circ }$

Possible Answers:

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.

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:

Each statement alone allows us to calculate the measure of one of the angles by subtracting the sum of the other two from 180.

From Statement 1:

$m\angle A + m\angle B + m\angle C = 180^{\circ }$

$120^{\circ } + m\angle C = 180^{\circ }$

$m\angle C = 60^{\circ }$

From Statement 2:

$m\angle A + m\angle B + m\angle C = 180^{\circ }$

$m\angle A + 110^{\circ } = 180^{\circ }$

$m\angle A =70^{\circ }$

Neither statement alone is enough to answer the question, since either statement leaves enough angle measurement to allow one of the other triangles to be right or obtuse. But the two statements together allow us to calculate $m\angle B$ :

$70 ^{\circ }+ m\angle B + 60^{\circ }= 180^{\circ }$ :

$m\angle B = 50 ^{\circ}$

This allows us to prove $\Delta ABC$ acute.

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GMAT Math : Geometry

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #2481 : Gmat Quantitative Reasoning

Example Question #12 : Dsq: Calculating The Perimeter Of An Acute / Obtuse Triangle

Example Question #12 : Acute / Obtuse Triangles

Example Question #1 : Dsq: Calculating Whether Acute / Obtuse Triangles Are Congruent

Example Question #1 : Dsq: Calculating Whether Acute / Obtuse Triangles Are Congruent

Example Question #21 : Triangles

Example Question #22 : Triangles

Example Question #3 : Dsq: Calculating An Angle In An Acute / Obtuse Triangle

Example Question #4 : Dsq: Calculating An Angle In An Acute / Obtuse Triangle

Example Question #5 : Dsq: Calculating An Angle In An Acute / Obtuse Triangle

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