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ISEE Upper Level Math : Rectangles

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Example Questions

Example Question #301 : Geometry

A >0 . Which is the greater quantity?

(a) The perimeter of the square on the coordinate plane with vertices (0,0), (A,0 ), (0,A), (A,A)

(b) The perimeter of the rectangle on the coordinate plane with vertices (0,0), (A+2,0 ), (0,A-2), (A+2,A-2)

Possible Answers:

(a) and (b) are equal.

It is impossible to tell from the information given.

(b) is greater.

(a) is greater.

Correct answer:

(a) and (b) are equal.

Explanation:

(a) The square has sidelength A - 0 = A , and therefore has perimeter .

(b) The rectangle has width A + 2 - 0 = A +2 and height A - 2 - 0 = A - 2 , so therefore has perimeter

$\left ( A + 2 \right ) + \left ( A - 2 \right ) + \left ( A + 2 \right ) + \left ( A - 2\right ) = A + A + A + A + 2- 2 + 2 - 2 = 4A$ .

The perimeters are the same.

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Example Question #1 : How To Find The Perimeter Of A Rectangle

Your geometry book has a rectangular front cover which is 12 inches by 8 inches.

What is the perimeter of your book cover?

Possible Answers:

80 in^2

42 in

40 in^2

40 in

Correct answer:

40 in

Explanation:

Your geometry book has a rectangular front cover which is 12 inches by 8 inches.

What is the perimeter of your book cover?

Perimeter is the distance around the outside of a shape.

Perimeter of a rectangle can be found via the following:

$P_{rectangle}=2l+2w$

Plug in our length and width to find our answer:

P=2(12in)+2(8in)=24in+16in=40in

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Example Question #82 : Quadrilaterals

Your aunt and uncle live on a rectangular plot of land out in the countryside. The width of their plot is $\frac{3}{4}$ of a kilometer long. The length is twice the width. What is the total length of the perimeter of the plot of land?

Possible Answers:

6 km

4.5 km

2.25 km

3.6 km

Correct answer:

4.5 km

Explanation:

Begin by finding both the width and length.

We are told that the width is $\frac{3}{4}$ of a kilometer

Further, we are told that the length is twice the width. Write this as an equation and solve:

$l=2w=2\frac{3}{4}=\frac{6}{4}=1\frac{1}{2}$

Now, find the perimeter via the following:

$P=2l+2w\rightarrow 2(\frac{3}{4}km)+ 2(\frac{3}{2}km)=1\frac{1}{2}km+3km=4.5km$

So, our perimeter is 4.5 km

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

The area of a rectangle is 100 square inches. What is the width of the rectangle?

Possible Answers:

$25\ in$

$20\ in$

Cannot be determined

$10\ in$

$4\ in$

Correct answer:

Cannot be determined

Explanation:

Without more information we cannot determine the width of a rectangle based on its area alone.

If we know this is a square and therefore that all sides are equal, we could simply take the square root of 100, which is 10.

However, we do not know that all sides are equal. We need to know the length of the rectangle to solve this problem. Therefore the best we can do is see what the equation would look like if we knew the length.

$w\times l=100$

$w=\frac{100}{l}$

where is the width and is the length.

Note: the incorrect answer choices are all factors of 100 and possible answers, but we can't be sure which, if any, are the right one.

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

The area of a rectangle is 40; its width is s + 5 . Which of the following expressions is equal to the length of the rectangle?

Possible Answers:

35 -s

$\frac{35}{s}$

$\frac{40}{s + 5}$

15 -s

Correct answer:

$\frac{40}{s + 5}$

Explanation:

Let be the length. The area of the rectangle is A = LW . Replace A = 40,W = s + 5 and solve for :

$L \left ( s+5 \right ) = 40$

$L \left ( s+5 \right ) \div \left ( s+5 \right ) = 40 \div \left ( s+5 \right )$

$L = \frac{40}{s + 5}$

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

The perimeter of a rectangle is 70; its width is t + 16 . Which of the following expressions is equal to the length of the rectangle?

Possible Answers:

54- t

$\frac{70}{t+16}$

$\frac{54}{t }$

19 - t

Correct answer:

19 - t

Explanation:

Let be the length. The perimeter of the rectangle is P = 2L + 2W .

Replace A = 70,W = t+16 and solve for :

P = 2L + 2W

$2L + 2 \left ( t+16 \right ) = 70$

2L + 2t+32 = 70

2L + 2t+32 -32 -2t = 70 -32 -2t

2L =38 -2t

$2L \div 2 =\left ( 38 -2t \right ) \div 2$

L = 19 - t

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

A rectangle has perimeter 56. Its length is 3x - 8 ; give its width in terms of .

Possible Answers:

-3x + 64

-3x +16

Insufficient information is given to answer the question.

-3x + 36

-3x +20

Correct answer:

-3x + 36

Explanation:

Substitute P = 56, L = 3x-8 in the formula for the perimeter of a rectangle:

2L + 2W = P

$2 \left ( 3x-8 \right ) + 2W = 56$

$2\cdot 3x-2\cdot 8 \right ) + 2W = 56$

6x-16+ 2W = 56

6x-6x-16+ 16 + 2W = -6x + 56+16

2W = -6x + 72

$2W \div 2 =\left ( -6x + 72 \right ) \div 2$

W = -3x + 36

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Example Question #312 : Geometry

You have a shed that you use for storing your gardening supplies and outdoor tools. If the shed has a footprint of and a width of 16 ft, what is the length of the shed? (assume the base is a rectangle)

Possible Answers:

17.75 ft

23 ft

24 ft

16.25 ft

Correct answer:

17.75 ft

Explanation:

We are given a rectangle with a known area and width, and asked to find the length.

To do so, recall this formula:

A=l*w

We know A and w, so simply work backwards to solve for l

$l=\frac{A}{w}=\frac{284ft^2}{16ft}=17.75ft$

So, our answer is 17.75 ft

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ISEE Upper Level Math : Rectangles

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

All ISEE Upper Level Math Resources

Example Questions

Example Question #301 : Geometry

Example Question #1 : How To Find The Perimeter Of A Rectangle

Example Question #82 : Quadrilaterals

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

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

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

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

Example Question #312 : Geometry

All ISEE Upper Level Math Resources

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