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GMAT Math : Quadrilaterals

Study concepts, example questions & explanations for GMAT Math

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Example Questions

Example Question #31 : Quadrilaterals

The area of a square that has sides with a length of 12 inches is equal to the area of a rectangle. If the rectangle has a width of 3 inches, what is the length of the rectangle?

Possible Answers:

\dpi{100} \small 48in.

\dpi{100} \small 40in.

\dpi{100} \small 24in.

\dpi{100} \small 36in.

\dpi{100} \small 12in.

Correct answer:

\dpi{100} \small 48in.

Explanation:

If the area of the rectangle is equal to the area of the square, then it must have an area of \dpi{100} \small 144in^{2} \dpi{100} \small (12\times 12). If the rectangle has an area of \dpi{100} \small 144 in^{2} and a side with a lenth of 3 inches, then the equation to solve the problem would be \dpi{100} \small 144=3x, where \dpi{100} \small x is the length of the rectangle. The solution:

\frac{144}{3} = 48\displaystyle \frac{144}{3} = 48.

Example Question #31 : Rectangles

A rectangle twice as long as it is wide has perimeter \displaystyle 6N - 12. Write its area in terms of \displaystyle N.

Possible Answers:

\displaystyle 2N^{2} -6N + 4

\displaystyle 4N^{2} -16N + 16

\displaystyle 2N^{2} -8N + 8

\displaystyle N^{2} -4N + 4

\displaystyle N^{2} -3N + 2

Correct answer:

\displaystyle 2N^{2} -8N + 8

Explanation:

Let \displaystyle W be the width of the rectangle; then its length is \displaystyle L = 2W, and its perimeter is 

\displaystyle 2W + 2 (2W) = 2W + 4W = 6W

Set this equal to \displaystyle 6N - 12 and solve for \displaystyle W:

\displaystyle 6W = 6N - 12

\displaystyle \frac{6W}{6} =\frac{ 6N-12}{6}

\displaystyle W = N - 2

The width is \displaystyle W = N - 2 and the length is  \displaystyle L = 2W = 2 \left (N - 2 \right ) = 2N - 4,  so multiply these expressions to get the area:

\displaystyle A = LW = (N-2) \cdot \left (2N-4 \right ) = 2N^{2} -8N + 8

Example Question #32 : Rectangles

A rectangle has its vertices at \displaystyle (-4,-3), (-4,7), (1,7), (1,-3). What part, in percent, of the rectangle is located in Quadrant III?

Possible Answers:

\displaystyle 12\%

\displaystyle 21 \%

\displaystyle 14 \%

\displaystyle 28 \%

\displaystyle 24 \%

Correct answer:

\displaystyle 24 \%

Explanation:

A rectangle with vertices \displaystyle (-4,-3), (-4,7), (1,7), (1,-3) has width \displaystyle 1 - (-4) = 5 and height \displaystyle 7 - (-3) = 10 , thereby having area \displaystyle 10 \times 5 = 50.

The portion of the rectangle in Quadrant III is a rectangle with vertices

\displaystyle (-4,-3), (-4,0), (0,0), (0,-3).

It has width \displaystyle 0-(-4) = 4 and height \displaystyle 0-(-3) = 3, thereby having area \displaystyle 4 \times 3 = 12 .

Therefore, \displaystyle \frac{12}{50 } of the rectangle is in Quadrant III; this is equal to 

\displaystyle \frac{12}{50 } \times 100 = 24 \%

Example Question #3 : Calculating The Area Of A Rectangle

What is the area of a rectangle given the length of \displaystyle 10 and width of \displaystyle 5?

 

Possible Answers:

\displaystyle 30

\displaystyle 50

\displaystyle 0.5

\displaystyle 15

Correct answer:

\displaystyle 50

Explanation:

To find the area of a rectangle, you must use the following formula:

\displaystyle A=lw

\displaystyle A=(10)(5)

\displaystyle A=50

Example Question #5 : Calculating The Area Of A Rectangle

What polynomial represents the area of a rectangle with length \displaystyle A + 4 and width \displaystyle A - 4 ?

Possible Answers:

\displaystyle A ^{2} -8A + 16

\displaystyle 4A

\displaystyle A ^{2} - 16

\displaystyle 2A

\displaystyle A ^{2} + 16

Correct answer:

\displaystyle A ^{2} - 16

Explanation:

The area of a rectangle is the product of the length and the width. The expression \displaystyle (A+4)(A-4) can be multplied by noting that this is the product of the sum and the difference of the same two terms; its product is the difference of the squares of the terms, or 

\displaystyle (A+4)(A-4) = A^{2}-4^{2}= A^{2}-16

Example Question #4 : Calculating The Area Of A Rectangle

A rectangle has its vertices at \displaystyle (-4,-3), (-4,7), (1,7), (1,-3). What percentage of the rectangle is located in Quadrant IV?

Possible Answers:

\displaystyle 3\%

\displaystyle 4 \%

\displaystyle 2 \%

\displaystyle 5\%

\displaystyle 6 \%

Correct answer:

\displaystyle 6 \%

Explanation:

A rectangle with vertices \displaystyle (-4,-3), (-4,7), (1,7), (1,-3) has width \displaystyle 1 - (-4) = 5 and height \displaystyle 7 - (-3) = 10 ; it follows that its area is \displaystyle 10 \times 5 = 50

The portion of the rectangle in Quadrant IV has vertices \displaystyle (0,0),(1,0), (1, -3), (0, -3). Its width is \displaystyle 1-0 = 1, and its height is \displaystyle 0- (-3 ) = 3, so its area is \displaystyle 1 \times 3 = 3.

Therefore, \displaystyle \frac{3}{50 }, or  \displaystyle \frac{3}{50} \times 100 \% = 6\%, of this rectangle is in Quadrant IV.

Example Question #6 : Calculating The Area Of A Rectangle

The perimeter of a rectangle is  and its length is \displaystyle 3 times the width. What is the area?

Possible Answers:

Correct answer:

Explanation:

The perimeter of a rectangle is the sum of all four sides, that is: \displaystyle 104= 2l + 2w

since \displaystyle l=3w, we can rewrite the equation as: 

\displaystyle 104= 2(3w) + 2w

\displaystyle 104= 8w

 

We are being asked for the area so we still aren't done. The area of a rectangle is the product of the width and length. We know what the width is so we can find the length and then take their product.

\displaystyle A= l \cdot w = 39 \cdot 13

Example Question #8 : Calculating The Area Of A Rectangle

Mark is building a garden with raised beds. One side of the garden will be 10 feet long and the other will be 5 less than three times the first side. What area of will Mark's garden be?

Possible Answers:

\displaystyle 300ft^2

\displaystyle 25ft^2

\displaystyle 250ft^2

\displaystyle 30ft^2

Correct answer:

\displaystyle 250ft^2

Explanation:

Mark is building a garden with raised beds. One side of the garden will be 10 feet long and the other will be 5 less than three times the first side. What area of will Mark's garden be?

This problem asks us to find the area of a rectangle. We are given one side and asked to find the other. To find the other, we need to use the provided clues.

"...five less..." \displaystyle -5

"...three times the first side..." \displaystyle 3x or \displaystyle 3(10)

So put it together:

\displaystyle 3(10)-5=30-5=25ft

Next, find the area via the following formula:

\displaystyle A=l*w=25ft*10ft=250ft^2

So our answer is:

\displaystyle 250ft^2

Example Question #5 : Calculating The Area Of A Rectangle

Find the area of a rectangle whose side lengths are .

Possible Answers:

\displaystyle 9

\displaystyle 63

\displaystyle 36

\displaystyle 7

Correct answer:

\displaystyle 63

Explanation:

To calculate area, multiply width times height. Thus,

\displaystyle 7*9=63

Example Question #6 : Calculating The Area Of A Rectangle

Find the area of a rectangle whose width is \displaystyle 2x and length is \displaystyle 2y.

Possible Answers:

\displaystyle 2x+2y

\displaystyle 4x+4y

\displaystyle 4xy

\displaystyle 2xy

Correct answer:

\displaystyle 4xy

Explanation:

To find area, simply multiply length times width. Thus

\displaystyle 2x\cdot2y=4xy

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