How to find the length of the side of a trapezoid - ISEE Upper Level: Quantitative Reasoning

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Question

In the above diagram, which depicts Trapezoid , and . Which is the greater quantity?

(a)

(b) 24

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Answer

To see that (b) is the greater quantity of the two, it suffices to construct the midsegment of the trapezoid - the segment which has as its endpoints the midpoints of legs $\overline{TP }$ and $\overline{RA }$ . Since and , the midsegment, $\overline{MN }$ , is positioned as follows:

The length of the midsegment is half the sum of the bases, so

$MN = \frac{1}{2} (TR + AP )= \frac{1}{2} \cdot (12 + 36 ) = 24$

, so .

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Question

Figure NOT drawn to scale.

The above figure depicts Trapezoid with midsegment $\overline{MN}$ . , and .

Give the area of Trapezoid in terms of .

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Answer

The midsegment of a trapezoid has as its length half the sum of the lengths of the bases, which here are $\overline{TR}$ and $\overline{AP }$ :

$MN = \frac{1}{2} (TR + AP )$

$20 = \frac{1}{2} (x+ y)$

$20 \cdot 2 = \frac{1}{2} (x+ y) \cdot 2$

Therefore,

The area of Trapezoid is one half multiplied by the height, , multiplied by the sum of the lengths of the bases, and . The midsegment of a trapezoid bisects both legs, so , and the area is

$A = \frac{1}{2} \cdot MP \cdot (MN + AP)$

$A = \frac{1}{2} \cdot 6 \cdot (20 + 40 - x )$

$A =3 \cdot (60 - x )$

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Question

The above figure depicts Trapezoid with midsegment $\overline{MN }$ . Express in terms of .

Tap to reveal answer

Answer

The midsegment of a trapezoid has as its length half the sum of the lengths of the bases, which here are $\overline{TR}$ and $\overline{AP }$ :

$MN = \frac{1}{2} (TR + AP )$

$20 = \frac{1}{2} (x+ y)$

$20 \cdot 2 = \frac{1}{2} (x+ y) \cdot 2$

The correct choice is .

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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

In the above diagram, which depicts Trapezoid , and . Which is the greater quantity?

(a)

(b) 24

Tap to reveal answer

Answer

To see that (b) is the greater quantity of the two, it suffices to construct the midsegment of the trapezoid - the segment which has as its endpoints the midpoints of legs $\overline{TP }$ and $\overline{RA }$ . Since and , the midsegment, $\overline{MN }$ , is positioned as follows:

The length of the midsegment is half the sum of the bases, so

$MN = \frac{1}{2} (TR + AP )= \frac{1}{2} \cdot (12 + 36 ) = 24$

, so .

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Question

Figure NOT drawn to scale.

The above figure depicts Trapezoid with midsegment $\overline{MN}$ . , and .

Give the area of Trapezoid in terms of .

Tap to reveal answer

Answer

The midsegment of a trapezoid has as its length half the sum of the lengths of the bases, which here are $\overline{TR}$ and $\overline{AP }$ :

$MN = \frac{1}{2} (TR + AP )$

$20 = \frac{1}{2} (x+ y)$

$20 \cdot 2 = \frac{1}{2} (x+ y) \cdot 2$

Therefore,

The area of Trapezoid is one half multiplied by the height, , multiplied by the sum of the lengths of the bases, and . The midsegment of a trapezoid bisects both legs, so , and the area is

$A = \frac{1}{2} \cdot MP \cdot (MN + AP)$

$A = \frac{1}{2} \cdot 6 \cdot (20 + 40 - x )$

$A =3 \cdot (60 - x )$

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Question

The above figure depicts Trapezoid with midsegment $\overline{MN }$ . Express in terms of .

Tap to reveal answer

Answer

The midsegment of a trapezoid has as its length half the sum of the lengths of the bases, which here are $\overline{TR}$ and $\overline{AP }$ :

$MN = \frac{1}{2} (TR + AP )$

$20 = \frac{1}{2} (x+ y)$

$20 \cdot 2 = \frac{1}{2} (x+ y) \cdot 2$

The correct choice is .

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Question

Note: Figure NOT drawn to scale.

The area of the above trapezoid is . What is ?

Tap to reveal answer

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

In the above diagram, which depicts Trapezoid , and . Which is the greater quantity?

(a)

(b) 24

Tap to reveal answer

Answer

To see that (b) is the greater quantity of the two, it suffices to construct the midsegment of the trapezoid - the segment which has as its endpoints the midpoints of legs $\overline{TP }$ and $\overline{RA }$ . Since and , the midsegment, $\overline{MN }$ , is positioned as follows:

The length of the midsegment is half the sum of the bases, so

$MN = \frac{1}{2} (TR + AP )= \frac{1}{2} \cdot (12 + 36 ) = 24$

, so .

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Question

Figure NOT drawn to scale.

The above figure depicts Trapezoid with midsegment $\overline{MN}$ . , and .

Give the area of Trapezoid in terms of .

Tap to reveal answer

Answer

The midsegment of a trapezoid has as its length half the sum of the lengths of the bases, which here are $\overline{TR}$ and $\overline{AP }$ :

$MN = \frac{1}{2} (TR + AP )$

$20 = \frac{1}{2} (x+ y)$

$20 \cdot 2 = \frac{1}{2} (x+ y) \cdot 2$

Therefore,

The area of Trapezoid is one half multiplied by the height, , multiplied by the sum of the lengths of the bases, and . The midsegment of a trapezoid bisects both legs, so , and the area is

$A = \frac{1}{2} \cdot MP \cdot (MN + AP)$

$A = \frac{1}{2} \cdot 6 \cdot (20 + 40 - x )$

$A =3 \cdot (60 - x )$

← Didn't Know|Knew It →

Question

The above figure depicts Trapezoid with midsegment $\overline{MN }$ . Express in terms of .

Tap to reveal answer

Answer

The midsegment of a trapezoid has as its length half the sum of the lengths of the bases, which here are $\overline{TR}$ and $\overline{AP }$ :

$MN = \frac{1}{2} (TR + AP )$

$20 = \frac{1}{2} (x+ y)$

$20 \cdot 2 = \frac{1}{2} (x+ y) \cdot 2$

The correct choice is .

← Didn't Know|Knew It →

Question

Note: Figure NOT drawn to scale.

The area of the above trapezoid is . What is ?

Tap to reveal answer

Answer

Substitute into the formula for the area of a trapezoid:

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

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

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

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

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

← Didn't Know|Knew It →

Question

In the above diagram, which depicts Trapezoid , and . Which is the greater quantity?

(a)

(b) 24

Tap to reveal answer

Answer

To see that (b) is the greater quantity of the two, it suffices to construct the midsegment of the trapezoid - the segment which has as its endpoints the midpoints of legs $\overline{TP }$ and $\overline{RA }$ . Since and , the midsegment, $\overline{MN }$ , is positioned as follows:

The length of the midsegment is half the sum of the bases, so

$MN = \frac{1}{2} (TR + AP )= \frac{1}{2} \cdot (12 + 36 ) = 24$

, so .

← Didn't Know|Knew It →

Question

Figure NOT drawn to scale.

The above figure depicts Trapezoid with midsegment $\overline{MN}$ . , and .

Give the area of Trapezoid in terms of .

Tap to reveal answer

Answer

The midsegment of a trapezoid has as its length half the sum of the lengths of the bases, which here are $\overline{TR}$ and $\overline{AP }$ :

$MN = \frac{1}{2} (TR + AP )$

$20 = \frac{1}{2} (x+ y)$

$20 \cdot 2 = \frac{1}{2} (x+ y) \cdot 2$

Therefore,

The area of Trapezoid is one half multiplied by the height, , multiplied by the sum of the lengths of the bases, and . The midsegment of a trapezoid bisects both legs, so , and the area is

$A = \frac{1}{2} \cdot MP \cdot (MN + AP)$

$A = \frac{1}{2} \cdot 6 \cdot (20 + 40 - x )$

$A =3 \cdot (60 - x )$

← Didn't Know|Knew It →

Question

The above figure depicts Trapezoid with midsegment $\overline{MN }$ . Express in terms of .

Tap to reveal answer

Answer

The midsegment of a trapezoid has as its length half the sum of the lengths of the bases, which here are $\overline{TR}$ and $\overline{AP }$ :

$MN = \frac{1}{2} (TR + AP )$

$20 = \frac{1}{2} (x+ y)$

$20 \cdot 2 = \frac{1}{2} (x+ y) \cdot 2$

The correct choice is .

← Didn't Know|Knew It →

Question

Note: Figure NOT drawn to scale.

The area of the above trapezoid is . What is ?

Tap to reveal answer

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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