GMAT Math : Problem-Solving Questions

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Example Questions

Example Question #73 : Triangles

Calculate the perimeter of an equilateral triangle whose side length is .

Possible Answers:

Correct answer:

Explanation:

To solve, simply multiply the side length by . Thus,

$P=3\cdot 3=9$

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Example Question #1 : Acute / Obtuse Triangles

A triangle has 2 sides length 5 and 12. Which of the following could be the perimeter of the triangle?

I. 20

II. 25

III. 30

Possible Answers:

I only

III only

II and III only.

All 3 are possible.

I and II only

Correct answer:

II and III only.

Explanation:

For a triangle, the sum of the two shortest sides must be greater than that of the longest. We are given two sides as 5 and 12. Our third side must be greater than 7, since if it were smaller than that we would have 5+x < 12 where is the unknown side. It must also be smaller than 17 since were it larger, we would have 5+12 < x .

Thus our perimeter will be between 5+7+12 = 24 and 5+12+17 = 34 . Only II and III are in this range.

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Example Question #2 : Calculating The Perimeter Of An Acute / Obtuse Triangle

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Triangle ABC has sides AB=3, BC=4, CA=5 . What is the perimeter of triangle ABC ?

Possible Answers:

Correct answer:

Explanation:

To calculate the perimeter, we simply need to add the three sides of the triangle.

Therefore, the perimeter is 3+4+5=12 , which is the final answer.

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Example Question #3 : Calculating The Perimeter Of An Acute / Obtuse Triangle

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Triangle ABC has height BD=4 . If is the midpoint of and DC=6 , what is the perimeter of triangle ABC ?

Possible Answers:

$2\sqrt{13}+12$

$2\sqrt{13}+13$

$4\sqrt{13}+12$

$4\sqrt{13}$

$2\sqrt{13}$

Correct answer:

$4\sqrt{13}+12$

Explanation:

Since BD is the height of triangle ABC, we can apply the Pythagorean Theorem to, let's say, triangle DBC and $BC=\sqrt{16+36}=2\sqrt{13}$ .

Since the basis of the height is at the midpoint of AC, it follows that triangle ABC, is an isoceles triangle.

We can find the perimeter by multiplying BC by 2 and add the basis of the triangle AC, which has length of $2\cdot6=12$ .

The final answer is therefore $4\sqrt{13}+12$ .

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Example Question #1 : Acute / Obtuse Triangles

An acute triangle has side lengths of , , and . What is the perimeter of the triangle?

Possible Answers:

11.5

Correct answer:

Explanation:

For any given triangle, the perimeter is the sum of the lengths of its sides. Given side lengths of , , and , P=7+12+4=23 .

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Example Question #4 : Calculating The Perimeter Of An Acute / Obtuse Triangle

An acute triangle has side lengths of , , and . What is the perimeter of the triangle?

Possible Answers:

Correct answer:

Explanation:

For any given triangle, the perimeter is the sum of the lengths of its sides. Given side lengths of , , and , P=5+11+6=22 .

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Example Question #3 : Acute / Obtuse Triangles

An acute triangle has side lengths of , , and . What is the perimeter of the triangle?

Possible Answers:

Correct answer:

Explanation:

For any given triangle, the perimeter is the sum of the lengths of its sides. Given side lengths of , , and , P=9+10+14=33 .

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

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

Suppose you want to answer this question, and you know that $\overline{AB} \cong \overline{DE}$ and $\overline{AC} \cong \overline{DF}$ . Which of the following additional facts would help you to answer this question one way or the other?

Possible Answers:

$\overline{BC} \cong \overline{EF}$

$\overline{AB} \cong \overline{DF}$

$\angle B\cong \angle E$

None of these statements would be sufficient to answer the question.

$\angle C\cong \angle F$

Correct answer:

$\overline{BC} \cong \overline{EF}$

Explanation:

If you know either that $\angle B\cong \angle E$ or $\angle C\cong \angle F$ , you have three congruencies - two sides and a nonincluded angle. This is not enough to establish triangle congruence,

If you know that $\overline{AB} \cong \overline{DF}$ , this, along with the other two statements, establishes that $\overline{AB} \cong \overline{AC} \cong \overline{DE} \cong \overline{DF}$ ; all this proves is that both triangles are isosceles.

If you also know that $\overline{BC} \cong \overline{EF}$ , however, the three statements together make three side congruencies, setting up the Side-Side-Side criterion for triangle congruence.

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Example Question #1 : Acute / Obtuse Triangles

Altitude

Note: Figure NOT drawn to scale

Refer to the above diagram. Which of the following statements is NOT a consequence of the fact that $\Delta ABX \cong \Delta ACX$ ?

Possible Answers:

$\Delta ABC$ is an equilateral triangle

$\angle AXC$ is a right angle

is the midpoint of $\overline{BC}$

$\overline{AX}$ is an altitude of $\Delta ABC$

$\overline{AX}$ bisects $\angle BAC$

Correct answer:

$\Delta ABC$ is an equilateral triangle

Explanation:

We use the congruence of corresponding sides and angles of congruent triangles to prove four of these statements:

$\angle CAX \cong \angle BAX$ , so $\overline{AX}$ bisects $\angle BAC$ .

$\overline{BX} \cong \overline{CX}$ , so is the midpoint of $\overline{BC}$ .

$\angle AXC \cong \angle AXB$ , and, since they form a linear pair, they are supplementary - therefore, they are right angles. Also, by definition, this makes $\overline{AX}$ an altitude.

However, nothing in this congruence proves that $\overline{BC}$ is congruent to the other two sides of $\Delta ABC$ (which are congruent). The correct statement to exclude is that $\Delta ABC$ is equilateral.

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Example Question #2 : Acute / Obtuse Triangles

Which of the following cannot be the measure of a base angle of an isosceles triangle?

Possible Answers:

$80 ^{\circ }$

Each of the other choices can be the measure of a base angle of an isosceles triangle.

$100 ^{\circ }$

$50^{\circ }$

$1 ^{\circ }$

Correct answer:

$100 ^{\circ }$

Explanation:

An isosceles triangle has two congruent angles by the Isosceles Triangle Theorem; these are the base angles. Since at least two angles of any triangle must be acute, a base angle must be acute - that is, it must measure under $90^{\circ }$ . The only choice that does not fit this criterion is $100 ^{\circ }$ , making this the correct choice.

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GMAT Math : Problem-Solving Questions

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #73 : Triangles

Example Question #1 : Acute / Obtuse Triangles

Example Question #2 : Calculating The Perimeter Of An Acute / Obtuse Triangle

Example Question #3 : Calculating The Perimeter Of An Acute / Obtuse Triangle

Example Question #1 : Acute / Obtuse Triangles

Example Question #4 : Calculating The Perimeter Of An Acute / Obtuse Triangle

Example Question #3 : Acute / Obtuse Triangles

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

Example Question #1 : Acute / Obtuse Triangles

Example Question #2 : Acute / Obtuse Triangles

All GMAT Math Resources

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