Quadrilaterals - GMAT Quantitative

Card 0 of 1104

Question

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?

Answer

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:

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

Compare your answer with the correct one above

Question

Note: figure NOT drawn to scale

Give the area of the above rectangle.

Answer

The area of a rectangle is the product of its length and width;

$A = 3^{x+1} \cdot 3^{x} = 3 ^{x+1+x} = 3 ^{2x+1}$

Compare your answer with the correct one above

Question

A rectangle twice as long as it is wide has perimeter . Write its area in terms of .

Answer

Let be the width of the rectangle; then its length is , and its perimeter is

Set this equal to and solve for :

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

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

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

Compare your answer with the correct one above

Question

A rectangle has its vertices at . What part, in percent, of the rectangle is located in Quadrant III?

Answer

A rectangle with vertices has width and height , thereby having area $10 \times 5 = 50$ .

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

.

It has width and height , thereby having area $4 \times 3 = 12$ .

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

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

Compare your answer with the correct one above

Question

A rectangle has its vertices at . What percentage of the rectangle is located in Quadrant IV?

Answer

A rectangle with vertices has width and height ; it follows that its area is $10 \times 5 = 50$ .

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

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

Compare your answer with the correct one above

Question

What is the area of a rectangle given the length of and width of ?

Answer

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

Compare your answer with the correct one above

Question

What polynomial represents the area of a rectangle with length and width ?

Answer

The area of a rectangle is the product of the length and the width. The expression 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

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

Compare your answer with the correct one above

Question

The perimeter of a rectangle is $104\textup{ inches}$ and its length is times the width. What is the area?

Answer

The perimeter of a rectangle is the sum of all four sides, that is:

since , we can rewrite the equation as:

$w=13\textup{ in}$

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.

$l=3w=3(13)=39\textup{ in}$

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

$A = 507\textup{ in}^2$

Compare your answer with the correct one above

Question

Find the area of a rectangle whose side lengths are $7\textup{ and }9$ .

Answer

To calculate area, multiply width times height. Thus,

Compare your answer with the correct one above

Question

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?

Answer

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..."

"...three times the first side..." or

So put it together:

Next, find the area via the following formula:

So our answer is:

Compare your answer with the correct one above

Question

Find the area of a rectangle whose width is and length is .

Answer

To find area, simply multiply length times width. Thus

$2x\cdot2y=4xy$

Compare your answer with the correct one above

Question

Write, in terms of , the perimeter of a square whose area is $N^{2 } - 6N + 9$

Answer

To find the perimeter of a square given its area, take the square root of the area to find its sidelength; then, multiply that sidelength by 4.

$N^{2 } - 6N + 9$ is a perfect square trinomial, since $\left (\frac{-6}{2} \right )^{2} =(-3)^{2}= 9$

so its square root is $\sqrt{N^{2 } - 6N + 9} = N - 3$ , the sidelength.

Multiply this by 4 to get the perimeter: $4 \left (N - 3 \right ) = 4N - 12$

Compare your answer with the correct one above

Question

If the diagonal of a square room is $\frac{x}{2}$ . What is the area of the room?

Answer

Cutting the triangle in half yields a right triangle with the diagonal becoming the hypotenuse and the other two legs being the sides of the square. Using the Pythagorean Theorem, we can solve for the other legs of the triangle.

Since both sides of the square are equal to eachother, , therefore:

$2a^2=\left(\frac{x}{2}\right)^2$

$2a^2= \frac{x^2}{4}$

$a^2=\frac{x^2}{8}$

$a= \frac{x}{\sqrt{8}}$

To find the area of the square:

with leg being one of the sides

$A= \left(\frac{x}{\sqrt{8}}\right)^2$

$A=\frac{x^2}{8}$

Compare your answer with the correct one above

Question

A square plot of land has perimeter 1,200 feet. Give its area in square yards.

Answer

The length of one side of the square is $1,200 \div 4 = 300$ feet, or $300 \div 3 = 100$ yards. Square this to get the area in square yards:

$A =100^{2} =10,000$ square yards.

Compare your answer with the correct one above

Question

What polynomial represents the area of Square if ?

Answer

As a square, is also a rhombus. The area of a rhombus is half the product of the lengths of its diagonals, one of which is $\overline{AC }$ . Since the diagonals are congruent, this is equal to half the square of :

$\frac{1}{2} \left (AC \right) ^{2}$

$= \frac{1}{2} \left (2t + 5 \right) ^{2}$

$= \frac{1}{2}\left [ \left (2t \right) ^{2} + 2\left (2t \right) \cdot 5 + 5^{2} \right ]$

$= \frac{1}{2}\left ( 4t ^{2} +20t + 25 \right )$

$= 2t ^{2} +10t +\frac{ 25 }{2}$

Compare your answer with the correct one above

Question

Six squares have sidelengths 8 inches, 1 foot, 15 inches, 20 inches, 2 feet, and 25 inches. What is the sum of their areas?

Answer

The areas of the squares are the squares of the sidelengths, so add the squares of the sidelengths. Since 1 foot is equal to 12 inches and 2 feet are equal to 24 inches, the sum of the areas is:

$8^{2} + 12 ^{2} + 15 ^{2} + 20 ^{2} + 24^{2} + 25^{2}$

square inches

Compare your answer with the correct one above

Question

A square, a regular pentagon, and a regular hexagon have the same sidelength. The sum of their perimeters is one mile. What is the area of the square?

Answer

The square, the pentagon, and the hexagon have a total of 15 sides, all of which are of equal length; the sum of the lengths is one mile, or 5,280 feet, so the length of one side of any of these polygons is

$5,280 \div 15 = 352$ feet.

The square has area equal to the square of this sidelength:

$352^{2} = 123,904 \textup{ ft} ^{2}$

Compare your answer with the correct one above

Question

A square and a regular pentagon have the same perimeter. The length of one side of the pentagon is 60 centimeters. What is the area of the square?

Answer

The regular perimeter has sidelength 60 centimeters and therefore perimeter $5 \times 60 = 300$ centimeters. The square has as its sidelength $300 \div 4 = 75$ centimeters and area $75 ^{2} = 5,625$ square centimeters.

Compare your answer with the correct one above

Question

Given square FGHI, answer the following

If square represents the surface of an ancient arena discovered by archaeologists, what is the area of the arena?

Answer

This problem requires us to find the area of a square. Don't let the story behind it distract you, it is simply an area problem. Use the following equation to find our answer:

is the length of one side of the square; in this case we are told that it is , so we can solve accordingly!

Compare your answer with the correct one above

Question

Note: Figure NOT drawn to scale

Refer to the above figure, which shows Square and Square . and Square has area 49. Give the area of Square .

Answer

Square has area 49, so each of its sides has as its length the square root of 49, or 7. Each side of Square is therefore a hypotenuse of a right triangle with legs 1 and , so each sidelength, including , can be found using the Pythagorean Theorem:

$WX = \sqrt{(WB)^{2}+(BX)^{2}}$

$WX = \sqrt{1^{2}+6^{2}} = \sqrt{1 +36} = \sqrt{37}$

The square of this, which is 37, is the area of Square .

Compare your answer with the correct one above

Tap the card to reveal the answer