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ISEE Upper Level Math : Trapezoids

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Example Questions

Example Question #1 : Trapezoids

The bigger base length of a trapezoid is two times bigger than the smaller base length. Legs of the trapezoid have the same length of $10 \ in$ . If the perimeter of the trapezoid is $56\ in$ , give the length of the smaller base.

Possible Answers:

$16\ in$

$15\ in$

$10\ in$

$12\ in$

$14\ in$

Correct answer:

$12\ in$

Explanation:

Like any polygon, the perimeter of a trapezoid is the total distance around the outside, i.e.

Perimeter=a+b+c+d ,

where a+b+c+d are the lengths of each side.

Let small base length and 2b= big base length.

$b+2b+10+10=56\Rightarrow 3b=36\Rightarrow b=12\ in$

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Example Question #2 : Trapezoids

The perimeter of the following trapezoid is $41\ cm$ .

The following is given:

a=2x

b=3x

c=1.5x

d=1.7x

Find .

Figure not drawn to scale.

Trapezoidr

Possible Answers:

Correct answer:

Explanation:

The perimeter of a trapezoid is equal to the sum of all of the sides, i.e.

Perimeter=a+b+c+d ,

where a,b,c,d are the lengths of each side.

$2x+3x+1.5x+1.7x=41\Rightarrow 8.2x=41\Rightarrow x=41\div 8.2=5$

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Example Question #1 : How To Find The Area Of A Trapezoid

Trapezoid

Note: Figure NOT drawn to scale.

The above trapezoid has area . What is ?

Possible Answers:

h = 10

h = 9

h = 12

h = 7

h = 14

Correct answer:

h = 7

Explanation:

Substitute A = 98, B = 16, b = 12 in the formula for the area of a trapezoid:

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

$\frac{1}{2} (16 + 12)h = 98$

$\frac{1}{2} (28)h = 98$

14h = 98

$14h \div 14= 98 \div 14$

h = 7

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Example Question #2 : How To Find The Area Of A Trapezoid

A trapezoid has height 20 inches. Its bases have sum 30 inches, and one base is 6 inches longer than the other. What is the area of this trapezoid?

Possible Answers:

$360 \textrm{ in}^{2}$

It cannot be determined from the information given.

$240 \textrm{ in}^{2}$

$300 \textrm{ in}^{2}$

$600 \textrm{ in}^{2}$

Correct answer:

$300 \textrm{ in}^{2}$

Explanation:

You do not need to find the individual bases; their sum, , is B + b . You can substitute B + b =30, h = 20 into the formula for the area of a trapezoid:

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

$A = \frac{1}{2}\cdot 30\cdot 20 = 300$ square inches.

Note that the fact that one base is inches longer is not important here.

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Example Question #265 : Geometry

Trapezoidr

Find the area of the above trapezoid if $a=12\ cm$ , $b=14\ cm$ , and $h=10\ cm$ .

Figure not drawn to scale.

Possible Answers:

$130\ cm^2$

$100\ cm^2$

$110\ cm^2$

$90\ cm^2$

$120\ cm^2$

Correct answer:

$130\ cm^2$

Explanation:

The area of a trapezoid is given by

$Area=(\frac{b_{1}+b_{2}}{2})h$ ,

where $b_{}1$ , $b_{}2$ are the lengths of each base and is the altitude (height) of the trapezoid.

$b_{1}=a=12\ cm$

$b_{2}=b=14\ cm$

$h=10\ cm$

$Area=(\frac{b_{1}+b_{2}}{2})h=(\frac{12+14}{2})\times 10=13\times 10=130\ cm^2$

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Example Question #2 : How To Find The Area Of A Trapezoid

A trapezoid has the base lengths of and . The area of the trapezoid is 8t^2 . Give the height of the trapezoid in terms of .

Possible Answers:

Correct answer:

Explanation:

The area of a trapezoid is given by

$Area=(\frac{b_{1}+b_{2}}{2})h$ ,

where $b_{}1$ , $b_{}2$ are the lengths of each base and is the altitude (height) of the trapezoid.

$Area=(\frac{b_{1}+b_{2}}{2})h=(\frac{3t+t}{2})\times h=8t^2\Rightarrow (2t)h=8t^2$

$\Rightarrow h=\frac{8t^2}{2t}=4t$

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Example Question #3 : How To Find The Area Of A Trapezoid

In the following trapezoid b=2a and a=h . The area of the trapezoid is 54 square inches. Give the height of the trapezoid. Figure not drawn to scale.

Trap

Possible Answers:

$4\ in$

$5\ in$

$7\ in$

$3\ in$

$6\ in$

Correct answer:

$6\ in$

Explanation:

The area of a trapezoid is given by

$Area=(\frac{b_{1}+b_{2}}{2})h$ ,

where $b_{}1$ , $b_{}2$ are the lengths of each base and is the altitude (height) of the trapezoid.

$b_{1}=a$

$b_{2}=b=2a$

h=a

Substitute these values into the area formula:

$Area=(\frac{b_{1}+b_{2}}{2})h=(\frac{a+2a}{2})a=54\Rightarrow \frac{3a^2}{2}=54\Rightarrow a^2=36\Rightarrow a=6\ in$

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Example Question #4 : How To Find The Area Of A Trapezoid

Square 3

What is the area of the shaded portion of the above square?

Possible Answers:

401

408

492

551

Correct answer:

492

Explanation:

Quadrilateral SQUR - the shaded region - is a trapezoid with bases $\overline{SR}$ and $\overline{QU }$ , and altitude $\overline{SQ}$ . The area of the trapezoid can be calculated using the formula

$A = \frac{1}{2} ( b_{1}+ b_{2} ) h$ ,

where $b_{1} = SR = 24$ and $b_{2} = QU$ , and h = SQ = 25 .

The length of $\overline{UA}$ can be found by setting c = 25 and b = 24 and applying the Pythagorean Theorem:

$a = \sqrt{c^{2} - b^{2}} = \sqrt{25^{2} - 24^{2}} =\sqrt{625- 576} = \sqrt{49} = 7$

Therefore,

QU = QA - UA = 24 - 7 = 17 .

Substituting:

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

$A = \frac{1}{2} ( 24 + 17) 24$

$A = \frac{1}{2} \cdot 41 \cdot 24$

A = 492

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Example Question #1 : How To Find The Length Of The Side Of A Trapezoid

Trapezoid

Note: Figure NOT drawn to scale.

The area of the above trapezoid is 465 . What is ?

Possible Answers:

B = 31

B = 47

B = 37

B = 42

B = 35

Correct answer:

B = 37

Explanation:

Substitute A = 465, b = 25, h = 15 into the formula for the area of a trapezoid:

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

$\frac{1}{2} (B + 25) \cdot 15 = 465$

$2 \cdot \frac{1}{2} (B + 25) \cdot 15 = 2 \cdot 465$

$(B + 25) \cdot 15 = 930$

15 B + 375 = 930

15 B + 375 - 375 = 930 - 375

15 B = 555

$15 B \div 15 = 555\div 15$

B = 37

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All ISEE Upper Level Math Resources

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ISEE Upper Level Math : Trapezoids

Study concepts, example questions & explanations for ISEE Upper Level Math

All ISEE Upper Level Math Resources

Example Questions

Example Question #1 : Trapezoids

Example Question #2 : Trapezoids

Example Question #1 : How To Find The Area Of A Trapezoid

Example Question #2 : How To Find The Area Of A Trapezoid

Example Question #265 : Geometry

Example Question #2 : How To Find The Area Of A Trapezoid

Example Question #3 : How To Find The Area Of A Trapezoid

Example Question #4 : How To Find The Area Of A Trapezoid

Example Question #1 : How To Find The Length Of The Side Of A Trapezoid

All ISEE Upper Level Math Resources

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