GMAT Math : Quadrilaterals

Study concepts, example questions & explanations for GMAT Math

varsity tutors app store varsity tutors android store

Example Questions

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:

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

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

\displaystyle 60 \textrm{ in}

\displaystyle 30 \textrm{ in}

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

Correct answer:

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

Explanation:

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

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

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

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

\displaystyle 12b^{2}= 6,000

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

\displaystyle b^{2} = 500

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

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

Example Question #231 : Geometry

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:

\displaystyle 2,640\textrm{ ft}

\displaystyle 4,640\textrm{ ft}

\displaystyle 3,960\textrm{ ft}

\displaystyle 4,500 \textrm{ ft}

\displaystyle 3,320\textrm{ ft}

Correct answer:

\displaystyle 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 

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

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

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

Possible Answers:

\displaystyle 25\:cm

\displaystyle 30\:cm

\displaystyle 50\:cm

\displaystyle 20\:cm

\displaystyle 40\:cm

Correct answer:

\displaystyle 30\:cm

Explanation:

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

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

\displaystyle 2A=(B+b)h

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

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

Plugging in our values for \displaystyle A, \displaystyle B, and \displaystyle h

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

\displaystyle 30=b 

Example Question #124 : Quadrilaterals

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

Possible Answers:

\displaystyle 40\:cm

\displaystyle 20\:cm

\displaystyle 50\:cm

\displaystyle 25\:cm

\displaystyle 45\:cm

Correct answer:

\displaystyle 40\:cm

Explanation:

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

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

 \displaystyle 2A=(B+b)h

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

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

Plugging in our values for \displaystyle A\displaystyle B, and \displaystyle h

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

\displaystyle 40=b

Example Question #125 : Quadrilaterals

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

Possible Answers:

\displaystyle 110\:cm

\displaystyle 80\:cm

\displaystyle 100\:cm

\displaystyle 95\:cm

\displaystyle 60\:cm

Correct answer:

\displaystyle 80\:cm

Explanation:

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

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

 \displaystyle 2A=(B+b)h

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

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

Plugging in our values for \displaystyle A\displaystyle B, and \displaystyle h

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

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

\displaystyle 200-120=b

\displaystyle 80=b

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 \displaystyle \textrm{Quad} \; ABCD is not a parallelogram, EXCEPT:

Possible Answers:

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

\displaystyle BC > 12

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

\displaystyle CD = 12

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

Correct answer:

\displaystyle BC > 12

Explanation:

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

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

Opposite angles of a parallelogram are congruent; if \displaystyle m \angle D < 76^{\circ}, then \displaystyle 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 \displaystyle BC > 12 would not by itself prove that the quadrilateral is not a parallelogram.

Example Question #2 : 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:

All four of the other choices fit the conditions.

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

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

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

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

Correct answer:

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

Explanation:

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

\displaystyle 75+85+95+105 = 360

\displaystyle 80+80+100+100 = 360

\displaystyle 60+100+100+100 = 360

\displaystyle 75 + 75 + 75 + 125 = 350

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

Example Question #2 : Calculating An Angle In A Quadrilateral

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

Possible Answers:

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

A right scalene triangle

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

A rectangle twice as long as it is wide

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

Correct answer:

A rhombus with a \displaystyle 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 \displaystyle 90^{\circ }; the rectangle fits this description, but the rhombus does not.

 

Example Question #2 : Calculating An Angle In A Quadrilateral

Rhombus \displaystyle RHOM has two diagonals that intersect at point \displaystyle P\displaystyle \angle RHO = 80 ^{\circ }

What is  \displaystyle m \angle RPH ?

Possible Answers:

\displaystyle 40 ^{\circ }

\displaystyle 50 ^{\circ }

\displaystyle 100 ^{\circ }

\displaystyle 90 ^{\circ }

\displaystyle 80 ^{\circ }

Correct answer:

\displaystyle 90 ^{\circ }

Explanation:

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

Example Question #1 : Calculating An Angle In A Quadrilateral

Quadrilateral \displaystyle QUAD is inscribed in circle \displaystyle \bigodot \displaystyle O\displaystyle m \angle A = 100 ^{\circ }. What is \displaystyle m \angle Q ?

Possible Answers:

\displaystyle 100 ^{\circ }

\displaystyle 50 ^{\circ }

\displaystyle 160 ^{\circ }

\displaystyle 80 ^{\circ }

\displaystyle 40^{\circ }

Correct answer:

\displaystyle 80 ^{\circ }

Explanation:

Two opposite angles of a quadrilateral inscribed inside a circle are supplementary, so 

\displaystyle m \angle Q = 180^{\circ } - m \angle A = 180^{\circ } - 100^{\circ } = 80^{\circ }

Tired of practice problems?

Try live online GMAT prep today.

1-on-1 Tutoring
Live Online Class
1-on-1 + Class
Learning Tools by Varsity Tutors