Trapezoids - ISEE Upper Level Quantitative Reasoning

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Question

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

The following is given:

Find .

Figure not drawn to scale.

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Answer

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

,

where 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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Question

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.

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Answer

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

,

where are the lengths of each side.

Let small base length and big base length.

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

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Question

Note: Figure NOT drawn to scale.

The area of the above trapezoid is . What is ?

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Answer

Substitute 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$

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

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

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Question

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

Figure not drawn to scale.

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Answer

The area of a trapezoid is given by

,

where , are the lengths of each base and is the altitude (height) of the trapezoid.

$h=10\ cm$

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Question

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

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Answer

The area of a trapezoid is given by

,

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

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

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Question

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

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Answer

The area of a trapezoid is given by

,

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

Substitute these values into the area formula:

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Question

Note: Figure NOT drawn to scale.

The above trapezoid has area . What is ?

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Answer

Substitute 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$

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

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Question

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?

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Answer

You do not need to find the individual bases; their sum, , is . You can substitute 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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Question

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

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Answer

Quadrilateral - 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 and , and .

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

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

Therefore,

.

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$

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Question

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

Figure not drawn to scale.

Tap to reveal answer

Answer

The area of a trapezoid is given by

,

where , are the lengths of each base and is the altitude (height) of the trapezoid.

$h=10\ cm$

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Question

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

Tap to reveal answer

Answer

The area of a trapezoid is given by

,

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

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

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Question

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

Tap to reveal answer

Answer

The area of a trapezoid is given by

,

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

Substitute these values into the area formula:

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Question

Note: Figure NOT drawn to scale.

The above trapezoid has area . What is ?

Tap to reveal answer

Answer

Substitute 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$

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

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Question

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?

Tap to reveal answer

Answer

You do not need to find the individual bases; their sum, , is . You can substitute 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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Question

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

Tap to reveal answer

Answer

Quadrilateral - 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 and , and .

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

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

Therefore,

.

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$

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Question

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

Figure not drawn to scale.

Tap to reveal answer

Answer

The area of a trapezoid is given by

,

where , are the lengths of each base and is the altitude (height) of the trapezoid.

$h=10\ cm$

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Question

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

Tap to reveal answer

Answer

The area of a trapezoid is given by

,

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

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

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Question

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

Tap to reveal answer

Answer

The area of a trapezoid is given by

,

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

Substitute these values into the area formula:

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Question

Note: Figure NOT drawn to scale.

The above trapezoid has area . What is ?

Tap to reveal answer

Answer

Substitute 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$

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

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Question

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?

Tap to reveal answer

Answer

You do not need to find the individual bases; their sum, , is . You can substitute 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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