Calculating the length of the side of a quadrilateral - GMAT Quantitative

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Question

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?

Answer

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

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

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

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

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

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Question

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?

Answer

Let be the length of the shorter base. Then is the length of its longer base, and is the height. Substitute 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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Question

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

Answer

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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Question

A given trapezoid has an area of , a longer base of length , and a height of . What is the length of the shorter base?

Answer

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 :

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

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

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

Plugging in our values for , , and :

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

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Question

A given trapezoid has an area of , a longer base of length , and a height of . What is the length of the shorter base?

Answer

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 :

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

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

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

Plugging in our values for , , and :

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

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Question

A given trapezoid has an area of , a longer base of length , and a height of . What is the length of the shorter base?

Answer

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 :

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

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

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

Plugging in our values for , , and :

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

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

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Question

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?

Answer

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

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

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

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

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

Compare your answer with the correct one above

Question

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?

Answer

Let be the length of the shorter base. Then is the length of its longer base, and is the height. Substitute 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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Question

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

Answer

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.

Compare your answer with the correct one above

Question

A given trapezoid has an area of , a longer base of length , and a height of . What is the length of the shorter base?

Answer

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 :

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

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

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

Plugging in our values for , , and :

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

Compare your answer with the correct one above

Question

A given trapezoid has an area of , a longer base of length , and a height of . What is the length of the shorter base?

Answer

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 :

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

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

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

Plugging in our values for , , and :

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

Compare your answer with the correct one above

Question

A given trapezoid has an area of , a longer base of length , and a height of . What is the length of the shorter base?

Answer

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 :

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

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

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

Plugging in our values for , , and :

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

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

Compare your answer with the correct one above

Question

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?

Answer

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

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

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

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

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

Compare your answer with the correct one above

Question

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?

Answer

Let be the length of the shorter base. Then is the length of its longer base, and is the height. Substitute 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.

Compare your answer with the correct one above

Question

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

Answer

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.

Compare your answer with the correct one above

Question

A given trapezoid has an area of , a longer base of length , and a height of . What is the length of the shorter base?

Answer

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 :

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

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

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

Plugging in our values for , , and :

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

Compare your answer with the correct one above

Question

A given trapezoid has an area of , a longer base of length , and a height of . What is the length of the shorter base?

Answer

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 :

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

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

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

Plugging in our values for , , and :

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

Compare your answer with the correct one above

Question

A given trapezoid has an area of , a longer base of length , and a height of . What is the length of the shorter base?

Answer

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 :

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

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

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

Plugging in our values for , , and :

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

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

Compare your answer with the correct one above

Question

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?

Answer

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

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

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

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

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

Compare your answer with the correct one above

Question

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?

Answer

Let be the length of the shorter base. Then is the length of its longer base, and is the height. Substitute 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.

Compare your answer with the correct one above

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