GMAT Math : Problem-Solving Questions

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Example Questions

Example Question #1 : Calculating An Angle In A Quadrilateral

Rectangle

Refer to the above figure. You are given that Polygon ABCD is a parallelogram, but NOT that it is a rectangle.

Which of the following statements is not enough to prove that Polygon ABCD is also a rectangle?

Possible Answers:

$\overline{BC} \perp \overline{DC}$

$m \angle 1 = 90^{\circ }$

$\angle BAC \cong \angle ACD$

$\angle BAC$ and $\angle CAD$ are complementary angles

$(AD)^{2} + (DC)^{2} = (AC)^{2}$

Correct answer:

$\angle BAC \cong \angle ACD$

Explanation:

To prove that Polygon ABCD is also a rectangle, we need to prove that any one of its angles is a right angle.

If $\overline{BC} \perp \overline{DC}$ , then by definition of perpendicular lines, $\angle BCD$ is right.

If $m \angle 1 = 90^{\circ }$ , then, since $\angle 1$ and $\angle BCD$ form a linear pair, $\angle BCD$ is right.

If $(AD)^{2} + (DC)^{2} = (AC)^{2}$ , then, by the Converse of the Pythagorean Theorem, $\Delta ADC$ is a right triangle with right angle $\angle D$ .

If $\angle BAC$ and $\angle CAD$ are complementary angles, then, since $m \angle BAC + m \angle CAD =m \angle BA D$

$m \angle BAD = m \angle BAC + m \angle CAD = 90 ^{\circ }$ , making $m \angle BAD$ right.

However, since, by definition of a parallelogram, $\overline{AB } \parallel \overline{ CD }$ , by the Alternate Interior Angles Theorem, $\angle BAC \cong \angle ACD$ regardless of whether the parallelogram is a rectangle or not.

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Example Question #3 : Calculating An Angle In A Quadrilateral

Two angles of a parallelogram measure $\left ( x + 70 \right )^{\circ }$ and $\left ( 2x\right +5) ^{\circ }$ . What are the possible values of ?

Possible Answers:

$x = 35 \textrm{ or } x = 65$

x = 65

$x = 35 \textrm{ or } x = 95$

x = 95

x = 35

Correct answer:

$x = 35 \textrm{ or } x = 65$

Explanation:

Case 1: The two angles are opposite angles of the parallelogram. In this case, they are congruent, and

2x +5 = x + 70

2x +5 - x - 5 = x + 70 - x - 5

x = 65

Case 2: The two angles are consecutive angles of the parallelogram. In this case, they are supplementary, and

$\left ( 2x +5 \right )+\left ( x + 70 \right ) = 180$

3x+ 75 = 180

3x+ 75 -75 = 180 -75

3x = 105

$3x \div 3 = 105\div 3$

x = 35

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Example Question #1 : Calculating The Perimeter Of A Quadrilateral

Quad

Note: Figure NOT drawn to scale

What is the perimeter of Quadrilateral QUAD , above?

Possible Answers:

Correct answer:

Explanation:

By the Pythagorean Theorem,

$\left (AD \right )^{2} +\left ( AU\right ) ^{2} =\left ( DU\right ) ^{2}$

$\left (AD \right )^{2} +4 ^{2} =5 ^{2}$

$\left (AD \right )^{2} +16 =25$

$\left (AD \right )^{2} +16 -16 =25 -16$

$\left (AD \right )^{2} =9$

AD = 3

Also by the Pythagorean Theorem,

$\left ( QU\right ) ^{2} + \left ( DU\right ) ^{2}=\left ( DQ\right ) ^{2}$

$\left ( QU\right ) ^{2} + 5^{2}=13^{2}$

$\left ( QU\right ) ^{2} + 25 =169$

$\left ( QU\right ) ^{2} + 25 -25=169 -25$

$\left ( QU\right ) ^{2} =144$

QU = 12

The perimeter of Quadrilateral QUAD is

QU + UA + AD + DQ = 12 + 4 + 3 + 13 = 32

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Example Question #131 : Quadrilaterals

What is the perimeter of Rhombus ABCD ?

Statement 1: AB = 10

Statement 2: Rhombus ABCD has area .

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.

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.

Correct answer:

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

Explanation:

A rhombus has four congruent sides. Statement 1 gives the length of one of them, so that length can be multiplied by to yield the perimeter.

The area of a rhombus alone has no bearing on its perimeter, so Statement 2 alone is insufficient.

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Example Question #245 : Geometry

Isosceles trapezoid FGHJ has a side length of $5 \:in$ . The length of one of its bases (Base A) is equal to three times the amount of seven inches fewer than the length of the other base (Base B). If Base B has a length of $17 \:in$ , what is the perimeter of FGHJ ?

Possible Answers:

$61 \:in$

$30 \:in$

$57 \:in$

$54 \:in$

$27 \:in$

Correct answer:

$57 \:in$

Explanation:

In an isosceles trapezoid, the two non-parallel sides are equal. In this case, they are both $5\:in$ long. To find the perimeter, we also need to know the length of both bases. We are told that Base B is $17 \:in$ . Eliminate any options less than or equal to $27 \:in$ .

To find the length of Base A, we need to translate the following: "The length of one of its bases (Base A) is equal to three times the amount of seven inches fewer than the length of the other base (Base B)." For translating, break the statement down:

Starting with $Base \:B=17 \:in$ ,

"seven inches fewer than the length of the other base (Base B)": $17\:in-7\:in$

"three times the amount of seven inches fewer than the length of the other base (Base B)": $3(17\:in-7\:in)$

$3(17\:in-7\:in)=3(10\:in)=30\:in$

So, Base A is $30 \:in$ long, so the perimeter of isosceles trapezoid FGHJ is:

$30\:in+5\:in+5\:in+17\:in=57 \:in$

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Example Question #2 : Calculating The Perimeter Of A Quadrilateral

Frank is planning on fencing a rectangular field near his house. The longer side of the house is two times four more than the length of the other side. If the shorter side is meters, what is the total length of fence that Frank needs?

Possible Answers:

176 m

116 m

6720 m

120 m

352 m

Correct answer:

352 m

Explanation:

This is a perimeter problem, but first we need to find our side lengths.

The short sides are meters. The long sides are two times 4 more than the short sides. So

"four more" 56+4=60

"two times" $60\cdot2=120$ meters

So our two side lengths are 120 and 56 meters. Find the perimeter by the following:

2(56)+2(120)=112+240=352

So, 352 meters

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Example Question #1 : Calculating The Perimeter Of A Quadrilateral

Export-png__4_

Rhombus ACBD has diagonals AB=4 and CD=2 . What is the perimeter of the rhombus?

Possible Answers:

$\sqrt{6}$

$2\sqrt{5}$

$4\sqrt{5}$

$\sqrt{5}$

Correct answer:

$4\sqrt{5}$

Explanation:

The rhombus is a special kind of a parallelogram. Its sides are all of the same length. Therefore, we just need to find one length of this quadrilateral. To do so, we can apply the Pythagorean Theorem on triangle AEC for example, since we know the length of the diagonals. Also, the diagonals intersect at their center. Therefore, triangle AEC has length, AE=2 and CE=1 . Therefore, $AC=\sqrt{1^{2}+2^{2}}$ or $AC= \sqrt{5}$ . The perimeter is then $4\sqrt{5}$ .

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Example Question #481 : Problem Solving Questions

Which set of side lengths cannot be the side lengths of a right triangle?

Possible Answers:

$\dpi{100} \small 48,64,80$

$\dpi{100} \small 84,35,91$

$\dpi{100} \small 28,45,53$

$\dpi{100} \small 45, 55,75$

$\dpi{100} \small 3,4,5$

Correct answer:

$\dpi{100} \small 45, 55,75$

Explanation:

For a triangle to be a right triangle, the sides must obey the Pythagorean Theorem. Let's try our options.

3, 4, 5: You should know this is a right triangle without having to do any calculations because it is one of the special triangles that you should remember. But if you didn't, $3^{2} + 4^{2} = 25 = 5^{2}$ .

28, 45, 53: $28^{2} + 45^{2} = 784 + 2025 = 2809 = 53^{2}$

45, 55, 75: $45^{2} + 55^{2} = 2025 + 3025 = 5050 \neq 75^{2}$ . The sides don't follow the Pythagorean Theorem so this can't be a right triangle. This is our answer. Let's check the remaining two sets of sides as well.

48, 64, 80: $48^{2} + 64^{2} = 2304 + 4096 = 6400 = 80^{2}$ . These are pretty big numbers and this math might take a while. Instead of doing these calculations, we could also see if 48, 64, and 80 look like any of the special triangles we know. Let's divide the three numbers by 16. 48/16 = 3, 64/16 = 4, and 80/16 = 5. Then this is just a type of 3,4,5 triangle, which we know is a right triangle.

84, 35, 91: $84^{2} + 35^{2} = 7056 + 1225 = 8281 = 91^{2}$ . Again, these are big numbers to square. Let's divide the three numbers by their greatest common factor, 7. 84/7 = 12, 35/7 = 5, 91/7 = 13. Then this is a 5, 12, 13 triangle, which is another of our special triangles that we know is a right triangle.

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Example Question #2 : Calculating Whether Right Triangles Are Similar

Which of the following right triangles is similar to one with a height of and a base of ?

Possible Answers:

h=8,b=14

h=12,b=20

h=5,b=10

h=7,b=12

h=10,b=15

Correct answer:

h=10,b=15

Explanation:

In order for two right triangles to be similar, the ratio of their dimensions must be equal. First we can check the ratio of the height to the base for the given triangle, and then we can check each answer choice for the triangle with the same ratio:

$Given:\frac{h}{b}=\frac{6}{9}=\frac{2}{3}$

So now we can check the ratio of the height to the base for each answer option, in no particular order, and the one with the same ratio as the given triangle will be a triangle that is similar:

$Option1:\frac{h}{b}=\frac{5}{10}=\frac{1}{2}$

$Option2:\frac{h}{b}=\frac{7}{12}$

$Option3:\frac{h}{b}=\frac{8}{14}=\frac{4}{7}$

$Option4:\frac{h}{b}=\frac{12}{20}=\frac{3}{5}$

$Option5:\frac{h}{b}=\frac{10}{15}=\frac{2}{3}$

The triangle with a height of and a base of has the same ratio as the given triangle, so this one is similar.

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Example Question #3 : Calculating Whether Right Triangles Are Similar

Which of the following right triangles is similar to one with a height of and a base of ?

Possible Answers:

h=3, b=5

h=7, b=14

h=6, b=2

h=2, b=6

h=3, b=1

Correct answer:

h=2, b=6

Explanation:

In order for two right triangles to be similar, their height-to-base ratios must be equal. Given a right triangle with a height h=4 and a base b=12 , the ratio $\frac{h}{b}=\frac{4}{12}=\frac{1}{3}$ . The only answer provided with a ratio of $\frac{1}{3}$ is h=2, b=6 .

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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 #1 : Calculating An Angle In A Quadrilateral

Example Question #3 : Calculating An Angle In A Quadrilateral

Example Question #1 : Calculating The Perimeter Of A Quadrilateral

Example Question #131 : Quadrilaterals

Example Question #245 : Geometry

Example Question #2 : Calculating The Perimeter Of A Quadrilateral

Example Question #1 : Calculating The Perimeter Of A Quadrilateral

Example Question #481 : Problem Solving Questions

Example Question #2 : Calculating Whether Right Triangles Are Similar

Example Question #3 : Calculating Whether Right Triangles Are Similar

All GMAT Math Resources

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