GMAT Math : Geometry

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Example Questions

Example Question #231 : Geometry

The area of a trapezoid is 1,600 square centimeters. Its height is 24 centimeters, and one base is one decimeter longer than the other. What is the length of the longer base?

Possible Answers:

$61 \frac{2}{3}\textrm{ cm}$

$71 \frac{2}{3}\textrm{ cm}$

$716 \frac{2}{3}\textrm{ cm}$

$616 \frac{2}{3}\textrm{ cm}$

$66 \frac{2}{3}\textrm{ cm}$

Correct answer:

$71 \frac{2}{3}\textrm{ cm}$

Explanation:

Let be the longer base. Then, since this is ten centimeters (= 1 decimeter) longer than the shorter, the shorter base has length B - 10 . Substitute A = 1,600, b = B - 10, h = 24 in the formula for the area of a trapezoid, and solve for :

$\frac{1}{2} (B + b)h = A$

$\frac{1}{2} (B + B - 10) \cdot 24= 1,600$

12 (2B - 10) = 1,600

24B - 120 = 1,600

24B = 1,720

$24B \div 24 = 1,720 \div 24$

$B = 71 \frac{2}{3}$ centimeters

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Example Question #1 : Calculating The Length Of The Side Of A Quadrilateral

The area of a trapezoid is 6,000 square inches. Its height is twice the length of its ionger base, which is three times the length of its shorter base. What is the height of the trapezoid?

Possible Answers:

$60 \sqrt{5 } \textrm{ in}$

$10 \sqrt{5 } \textrm{ in}$

$60 \textrm{ in}$

$30 \textrm{ in}$

$30 \sqrt{5 } \textrm{ in}$

Correct answer:

$60 \sqrt{5 } \textrm{ in}$

Explanation:

Let be the length of the shorter base. Then is the length of its longer base, and 2 (3b)= 6b is the height. Substitute B = 3b, h = 6b , A = 6,666 in the area formula:

$\frac{1}{2} ( B + b) h = A$

$\frac{1}{2} ( 3b + b) \cdot 6b= 6,000$

$\frac{1}{2} \cdot 4b \cdot 6b= 6,000$

$12b^{2}= 6,000$

$12b^{2} \div 12 = 6,000 \div 12$

$b^{2} = 500$

$b = \sqrt{500} = \sqrt{100}\cdot \sqrt{5 } =10 \sqrt{5 }$

The height is six times this, or $h = 6 \cdot 10 \sqrt{5 } = 60 \sqrt{5 }$ inches.

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Example Question #1 : Calculating The Length Of The Side Of A Quadrilateral

A trapezoid has bases of length one mile and 4,000 feet and height one-half mile. What is the length of its midsegment?

Possible Answers:

$2,640\textrm{ ft}$

$3,960\textrm{ ft}$

$3,320\textrm{ ft}$

$4,500 \textrm{ ft}$

$4,640\textrm{ ft}$

Correct answer:

$4,640\textrm{ ft}$

Explanation:

The length of the midsegment of a trapezoid is the mean of the lengths of its bases; the height is irrelevant. The bases are of length 4,000 feet and 5,280 feet; their mean is

$\frac{1}{2} \left ( 5,280 + 4,000 \right ) = 4,640$ feet, the length of the midsegment.

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Example Question #3 : Calculating The Length Of The Side Of A Quadrilateral

A given trapezoid has an area of $2500\:cm^2$ , a longer base of length $70\:cm$ , and a height of $50\:cm$ . What is the length of the shorter base?

Possible Answers:

$25\:cm$

$30\:cm$

$50\:cm$

$20\:cm$

$40\:cm$

Correct answer:

$30\:cm$

Explanation:

The area of a trapezoid with a larger base , a shorter base , and a height is defined by the equation $A=\frac{1}{2}(B+b)h$ . Restructuring to solve for the shorter base :

$A=\frac{1}{2}(B+b)h$

2A=(B+b)h

$\frac{2A}{h}=B+b$

$\frac{2A}{h}-B=b$

Plugging in our values for , , and :

$\frac{2(2500)}{50}-70=b$

30=b

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Example Question #124 : Quadrilaterals

A given trapezoid has an area of $1000\:cm^2$ , a longer base of length $60\:cm$ , and a height of $20\:cm$ . What is the length of the shorter base?

Possible Answers:

$40\:cm$

$20\:cm$

$50\:cm$

$25\:cm$

$45\:cm$

Correct answer:

$40\:cm$

Explanation:

The area of a trapezoid with a larger base , a shorter base , and a height is defined by the equation $A=\frac{1}{2}(B+b)h$ . Restructuring to solve for the shorter base :

$A=\frac{1}{2}(B+b)h$

2A=(B+b)h

$\frac{2A}{h}=B+b$

$\frac{2A}{h}-B=b$

Plugging in our values for , , and :

$\frac{2(1000)}{20}-60=b$

40=b

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Example Question #125 : Quadrilaterals

A given trapezoid has an area of $4000\:cm^2$ , a longer base of length $120\:cm$ , and a height of $40\:cm$ . What is the length of the shorter base?

Possible Answers:

$110\:cm$

$80\:cm$

$100\:cm$

$95\:cm$

$60\:cm$

Correct answer:

$80\:cm$

Explanation:

The area of a trapezoid with a larger base , a shorter base , and a height is defined by the equation $A=\frac{1}{2}(B+b)h$ . Restructuring to solve for the shorter base :

$A=\frac{1}{2}(B+b)h$

2A=(B+b)h

$\frac{2A}{h}=B+b$

$\frac{2A}{h}-B=b$

Plugging in our values for , , and :

$\frac{2(4000)}{40}-120=b$

$\frac{8000}{40}-120=b$

200-120=b

80=b

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Example Question #1 : Calculating An Angle In A Quadrilateral

Parallelogram

Note: Diagram is NOT drawn to scale.

Refer to the above diagram.

Any of the following facts alone would be enough to prove that $\textrm{Quad} \; ABCD$ is not a parallelogram, EXCEPT:

Possible Answers:

Any one of these facts alone would prove that $\textrm{Quad} \; ABCD$ is not a parallelogram.

$m \angle D <76^{\circ}$

CD = 12

BC > 12

$m \angle A = 114^{\circ}$

Correct answer:

BC > 12

Explanation:

Opposite sides of a parallelogram are congruent; if CD = 12 , then $CD \neq AB$ , violating this condition.

Consecutive angles of a parallelogram are supplementary; if $m \angle A = 114^{\circ}$ , then $m \angle A + m \angle B= 114 + 76 = 190$ , violating this condition.

Opposite angles of a parallelogram are congruent; if $m \angle D <76^{\circ}$ , then $m \angle D \neq m \angle B$ , violating this condition.

Adjacent sides of a parallelogram, however, may or may not be congruent, so the condition that BC > 12 would not by itself prove that the quadrilateral is not a parallelogram.

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Example Question #1 : Calculating An Angle In A Quadrilateral

Which of the following can not be the measures of the four interior angles of a quadrilateral?

Possible Answers:

$60^{\circ },100^{\circ },100^{\circ },100^{\circ }$

All four of the other choices fit the conditions.

$75^{\circ },75^{\circ },75^{\circ },125^{\circ }$

$80^{\circ },80^{\circ },100^{\circ },100^{\circ }$

$75^{\circ },85^{\circ },95^{\circ },105^{\circ }$

Correct answer:

$75^{\circ },75^{\circ },75^{\circ },125^{\circ }$

Explanation:

The four interior angles of a quadrilateral measure a total of $360^{\circ }$ , so we test each group of numbers to see if they have this sum.

75+85+95+105 = 360

80+80+100+100 = 360

60+100+100+100 = 360

75 + 75 + 75 + 125 = 350

This last group does not have the correct sum, so it is the correct choice.

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Example Question #3 : Calculating An Angle In A Quadrilateral

A circle can be circumscribed about each of the following figures except:

Possible Answers:

A right scalene triangle

A rhombus with a $60^{\circ }$ angle

A rectangle twice as long as it is wide

An isosceles trapezoid with one base three times as long as the other

An isosceles triangle with its base one-half as long as either leg

Correct answer:

A rhombus with a $60^{\circ }$ angle

Explanation:

A circle can be circumscribed about any triangle regardless of its sidelengths or angle measures, so we can eliminate the two triangle choices.

A circle can be circumscribed about a quadrilateral if and only if both pairs of its opposite angles are supplementary. An isosceles trapezoid has this characteristic; this can be proved by the fact that base angles are congruent, and by the Same-Side Interior Angles Statement. For a parallelogram to have this characteristic, since opposite angles are congruent also, all angles must measure $90^{\circ }$ ; the rectangle fits this description, but the rhombus does not.

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Example Question #4 : Calculating An Angle In A Quadrilateral

Rhombus RHOM has two diagonals that intersect at point ; $\angle RHO = 80 ^{\circ }$ .

What is $m \angle RPH$ ?

Possible Answers:

$40 ^{\circ }$

$100 ^{\circ }$

$90 ^{\circ }$

$80 ^{\circ }$

$50 ^{\circ }$

Correct answer:

$90 ^{\circ }$

Explanation:

The diagonals of a rhombus always intersect at right angles, so $m \angle HPO = 90 ^{\circ }$ . The measures of the interior angles of the rhombus are irrelevant.

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GMAT Math : Geometry

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #231 : Geometry

Example Question #1 : Calculating The Length Of The Side Of A Quadrilateral

Example Question #1 : Calculating The Length Of The Side Of A Quadrilateral

Example Question #3 : Calculating The Length Of The Side Of A Quadrilateral

Example Question #124 : Quadrilaterals

Example Question #125 : Quadrilaterals

Example Question #1 : Calculating An Angle In A Quadrilateral

Example Question #1 : Calculating An Angle In A Quadrilateral

Example Question #3 : Calculating An Angle In A Quadrilateral

Example Question #4 : Calculating An Angle In A Quadrilateral

All GMAT Math Resources

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