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

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Example Questions

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

Trapezoid

In the above diagram, which depicts Trapezoid TRAP , TX = RY = 14 and XP = YA = 15 . Which is the greater quantity?

(a)

(b) 24

Possible Answers:

(a) is the greater quantity

It is impossible to determine which is greater from the information given

(a) and (b) are equal

(b) is the greater quantity

Correct answer:

(b) is the greater quantity

Explanation:

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 TX < XP and RY < YA , the midsegment, $\overline{MN }$ , is positioned as follows:

Trapezoid

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

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

XY < MN , so XY < 24 .

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

Trapezoid 1

Figure NOT drawn to scale.

The above figure depicts Trapezoid TRAP with midsegment $\overline{MN}$ . MT = 6 , and RN = 9 .

Give the area of Trapezoid MNAP in terms of .

Possible Answers:

3x- 60

- 3x+ 180

3x + 180

3x+ 60

Correct answer:

- 3x+ 180

Explanation:

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

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

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

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

40 = x+ y

40 - x = y

Therefore, AP = 40 - x

The area of Trapezoid MNAP 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 MP = MT = 6 , and the area is

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

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

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

A =180 - 3x

A = - 3x+ 180

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

Trapezoid 1

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

Possible Answers:

y = 40 + x

y = 2x- 20

y = 2 x + 20

y = 40 - x

Correct answer:

y = 40 - x

Explanation:

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

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

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

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

40 = x+ y

40 - x = x+ y - x

40 - x = y

The correct choice is y = 40 - x .

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

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

All ISEE Upper Level Quantitative Resources

Example Questions

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

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

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

All ISEE Upper Level Quantitative Resources

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