GMAT Math : Calculating the length of the side of a quadrilateral

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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 #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:

$2,640\textrm{ ft}$

$3,320\textrm{ ft}$

$4,500 \textrm{ ft}$

$4,640\textrm{ ft}$

$3,960\textrm{ ft}$

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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GMAT Math : Calculating the length of the side of a quadrilateral

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 #231 : Geometry

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

Example Question #124 : Quadrilaterals

Example Question #125 : Quadrilaterals

All GMAT Math Resources

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