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

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Example Questions

Example Question #31 : Other Quadrilaterals

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:

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

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

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

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

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

Correct answer:

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

Explanation:

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

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

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

$\displaystyle 12 (2B - 10) = 1,600$

$\displaystyle 24B - 120 = 1,600$

$\displaystyle 24B = 1,720$

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

$B = 71 \frac{2}{3}$ $\displaystyle B = 71 \frac{2}{3}$ 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:

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

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

$60 \textrm{ in}$ $\displaystyle 60 \textrm{ in}$

$30 \textrm{ in}$ $\displaystyle 30 \textrm{ in}$

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

Correct answer:

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

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:

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

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

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

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

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

$b^{2} = 500$ $\displaystyle b^{2} = 500$

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

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

$2,640\textrm{ ft}$ $\displaystyle 2,640\textrm{ ft}$

$4,640\textrm{ ft}$ $\displaystyle 4,640\textrm{ ft}$

$3,960\textrm{ ft}$ $\displaystyle 3,960\textrm{ ft}$

$4,500 \textrm{ ft}$ $\displaystyle 4,500 \textrm{ ft}$

$3,320\textrm{ ft}$ $\displaystyle 3,320\textrm{ ft}$

Correct answer:

$4,640\textrm{ ft}$ $\displaystyle 4,640\textrm{ ft}$

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

$\frac{1}{2} \left ( 5,280 + 4,000 \right ) = 4,640$ $\displaystyle \frac{1}{2} \left ( 5,280 + 4,000 \right ) = 4,640$ 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 $2500\:cm^2$ $\displaystyle 2500\:cm^2$, a longer base of length $70\:cm$ $\displaystyle 70\:cm$ , and a height of $50\:cm$ $\displaystyle 50\:cm$. What is the length of the shorter base?

Possible Answers:

$25\:cm$ $\displaystyle 25\:cm$

$30\:cm$ $\displaystyle 30\:cm$

$50\:cm$ $\displaystyle 50\:cm$

$20\:cm$ $\displaystyle 20\:cm$

$40\:cm$ $\displaystyle 40\:cm$

Correct answer:

$30\:cm$ $\displaystyle 30\:cm$

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 $A=\frac{1}{2}(B+b)h$ $\displaystyle A=\frac{1}{2}(B+b)h$. Restructuring to solve for the shorter base $\displaystyle b$:

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

$\displaystyle 2A=(B+b)h$

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

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

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

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

30=b $\displaystyle 30=b$

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Example Question #124 : Quadrilaterals

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

Possible Answers:

$40\:cm$ $\displaystyle 40\:cm$

$20\:cm$ $\displaystyle 20\:cm$

$50\:cm$ $\displaystyle 50\:cm$

$25\:cm$ $\displaystyle 25\:cm$

$45\:cm$ $\displaystyle 45\:cm$

Correct answer:

$40\:cm$ $\displaystyle 40\:cm$

Explanation:

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

$\displaystyle 2A=(B+b)h$

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

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

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

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

40=b $\displaystyle 40=b$

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Example Question #125 : Quadrilaterals

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

Possible Answers:

$110\:cm$ $\displaystyle 110\:cm$

$80\:cm$ $\displaystyle 80\:cm$

$100\:cm$ $\displaystyle 100\:cm$

$95\:cm$ $\displaystyle 95\:cm$

$60\:cm$ $\displaystyle 60\:cm$

Correct answer:

$80\:cm$ $\displaystyle 80\:cm$

Explanation:

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

$\displaystyle 2A=(B+b)h$

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

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

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

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

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

$\displaystyle 200-120=b$

80=b $\displaystyle 80=b$

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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 #31 : Other Quadrilaterals

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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