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

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 .

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 .

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 ?

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

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 .

Compare your answer with the correct one above

Question

Figure NOT drawn to scale.

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

Give the area of Trapezoid in terms of .

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

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

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

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

Compare your answer with the correct one above

Question

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

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 .

Compare your answer with the correct one above

Question

Note: Figure NOT drawn to scale.

The area of the above trapezoid is . What is ?

Answer

Substitute into the formula for the area of a trapezoid:

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

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

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

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

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

Compare your answer with the correct one above

Question

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

(a)

(b) 24

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 .

Compare your answer with the correct one above

Question

Figure NOT drawn to scale.

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

Give the area of Trapezoid in terms of .

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

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

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

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

Compare your answer with the correct one above

Question

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

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 .

Compare your answer with the correct one above

Question

Note: Figure NOT drawn to scale.

The area of the above trapezoid is . What is ?

Answer

Substitute into the formula for the area of a trapezoid:

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

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

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

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

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

Compare your answer with the correct one above

Question

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

(a)

(b) 24

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 .

Compare your answer with the correct one above

Question

Figure NOT drawn to scale.

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

Give the area of Trapezoid in terms of .

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

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

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

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

Compare your answer with the correct one above

Question

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

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 .

Compare your answer with the correct one above

Question

Note: Figure NOT drawn to scale.

The area of the above trapezoid is . What is ?

Answer

Substitute into the formula for the area of a trapezoid:

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

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

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

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

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

Compare your answer with the correct one above

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