Squares - GMAT Quantitative Reasoning

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Question

If the perimeter of a square is , what is its area?

Answer

The perimeter of a square, and any shape for that matter, is found by adding up all the exterior sides. Since all sides are equal in a square, we can say:

where represents the length of a side

We can solve for the side length using the information provided:

The area of a square is found by squaring the side length:

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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: $4 \left (N - 3 \right ) = 4N - 12$

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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

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

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

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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.

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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 :

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

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

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

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

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

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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

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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:

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

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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.

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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!

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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 .

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Question

Note: Figure NOT drawn to scale

Refer to the above figure, which shows Square and Square . and Square has area 25. Give the area of Square .

Answer

Square has area 25, so each side has length the square root of 25, or 5.

Specifically, , and, as given, .

Since $\bigtriangleup WBX$ is a right triangle with hypotenuse $\overline{WX}$ and legs $\overline{BW}$ and $\overline{BX}$ , can be found using the Pythagorean Theorem:

$BX = \sqrt{(WX)^{2}-(BW)^{2}}$

$= \sqrt{5^{2}-1^{2}}$

$= \sqrt{25-1}$

$= \sqrt{24}$

$=\sqrt{4} \cdot \sqrt{6}$

$= 2\sqrt{6}$

The area of $\bigtriangleup WBX$ is

$\frac{1}{2} \cdot BW \cdot BX = \frac{1}{2} \cdot 1 \cdot 2 \sqrt{6} = \sqrt{6}$

Since all four triangles, by symmetry, are congruent, all have this area. the area of Square is the area of Square plus the areas of the four triangles, or $25 + 4\sqrt{6}$ .

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Question

Note: Figure NOT drawn to scale

Refer to the above figure, which shows Square and Square . The ratio of to is 13 to 2. What is the ratio of the area of Square to that of Square ?

Answer

To make this easier, assume that and - the reasoning generalizes. Then Square has sidelength 15 and area $15^{2}= 225$ . The sidelength of Square , each side being a hypotenuse of a right triangle with legs 2 and 13, is

$\sqrt{2^{2}+13^{2}}= \sqrt{4+169} = \sqrt{173}$ .

The square of this, 173, is the area of Square .

The ratio is .

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Question

Note: Figure NOT drawn to scale

Refer to the above figure, which shows Square and Square . The ratio of to is 7 to 1.

Which of these responses comes closest to what percent the area of Square is of that of Square ?

Answer

To make this easier, assume that and ; the results generalize.

Each side of Square has length 8, so the area of Square is 64.

Each of the four right triangles has legs 7 and 1, so each has area $\frac{1}{2} \cdot 1 \cdot 7 = 3\frac{1}{2}$ ; Square has area four times this subtracted from the area of Square , or

$64 - 4 \cdot 3 \frac{1}{2} = 50$ .

The area of Square is

$\frac{50}{64} \times 100 % = 78 \frac{1}{8} %$

of that of Square .

Of the five choices, 80% comes closest.

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Question

The perimeter of a square is the same as the circumference of a circle with area 100. What is the area of the square?

Answer

The formula for the area of a circle is

$A = \pi r^{2}$ .

If the area is 100, the radius is as follows:

$\pi r^{2} = 100$

$r^{2} = \frac{100}{\pi}$

$r = \sqrt{\frac{100}{\pi}}= \frac{10}{\sqrt{\pi}}$

The circle has circumference $2 \pi$ times its radius, or

$C= \frac{10}{\sqrt{\pi}} \cdot 2 \pi = 20 \sqrt{\pi}$

This is also the perimeter of the square, so the sidelength of the square is one-fourth this, or

$S = \frac{1}{4} \cdot 20 \sqrt{\pi} = 5 \sqrt{\pi}$

The area of the square is the square of this, or

$A'= \left (5 \sqrt{\pi} \right )^{2}= 25 \pi$

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Question

The perimeter of a square is the same as the circumference of a circle with radius 8. What is the area of the square?

Answer

A circle with radius 8 has as its circumference $2 \pi$ times this, or

$2 \pi \cdot 8 = 16 \pi$ .

This is also the perimeter of the square, so the sidelength is one fourth of this, or

$\frac{1}{4} \cdot 16 \pi = 4 \pi$ .

The area is the square of this, or

$\left ( 4 \pi \right )^{2} = 16 \pi^{2}$ .

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Question

The perimeter of a square is the same as the length of the hypotenuse of a right triangle with legs 8 and 12. What is the area of the square?

Answer

The length of the hypotenuse of a right triangle with legs 8 and 12 can be determined using the Pythagorean Theorem:

$\sqrt{8^{2}+12^{2}} = \sqrt{64+144}= \sqrt{208} = \sqrt{16} \cdot \sqrt{13} = 4\sqrt{13}$

Since this is also the perimeter of the square, its sidelength is one fourth of this, or

$\frac{ 4\sqrt{13} }{4} =\sqrt{13}$

The area of the square is the square of this sidelength, or

$\left ( \sqrt{13} \right )^{2}= 13$

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Question

The perimeter of a square is $4 ^{d}$ . Give its area.

Answer

The length of one side of a square is the perimeter divided by 4:

$\frac{4 ^{d} }{4} = \frac{4 ^{d} }{4 ^{1}} = 4 ^{d-1}$

Square this to get the area:

$( 4 ^{d-1} )^{2} = 4 ^{(d-1) \cdot 2 } = 4 ^{2d- 2 }$

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Question

A square has a side length of 3, how long is the diagonal of the square?

Answer

The diagonal divides the square into two 45-45-90 triangles, which have side lengths in the ratio of $1:1:\sqrt2$ . Therefore, the diagonal is the side length $x\sqrt2$ .

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Question

Given square FGHI below, solve for the following:

If square represents a grassy area in the middle of your college campus, what is the shortest distance from point to point ?

Answer

There are two pretty easy ways to solve this problem, both of which require you to see that this is actually a triangle question.

The first method is the fastest. If you can recognize that the diagonal of a square makes a triangle, you can then recall that triangles have side lengths with ratios of $1:1:\sqrt2$ . Therefore:

$Diagonal:Length=15\sqrt{2} =15\sqrt{2}:m$

The second way to solve this problem is to use Pythagorean Theorem:

where and are leg lengths of a triangle and the length of the hypotenuse. In this case, we want to solve for the diagonal, which is the hypotenuse of the triangle it forms, so we are solving for :

$c=\sqrt{a^2+b^2}=\sqrt{15^2+15^2}=\sqrt{450}=\sqrt{2225}+\sqrt{225}*\sqrt{2}=15\sqrt{2}$

So again, we get $15\sqrt{2} :m$ .

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Question

One side of a square has a length of . What is the length of the diagonal of the square?

Answer

By definition, all four sides of a square are the same length. So if one side has a length of , the length and width of the square are both . The diagonal of the square is just the hypotenuse of a right triangle with the length and width as the other two sides, so we can use the Pythagorean Theorem to calculate the length of the diagonal:

$d=\sqrt{98}=\sqrt{2\cdot 49}=7\sqrt{2}$

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