Squares - ISEE Upper Level Mathematics Achievement

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Question

A square is made into a rectangle by increasing the width by 20% and decreasing the length by 20%. By what percentage has the area of the square changed?

Answer

The area decreases by 20% of 20%, which is 4%.

The easiest way to see this is to plug in numbers for the sides of the square. If we are using percentages, it is easiest to use factors of 10 or 100. In this case we will say that the square has a side length of 10.

10% of 10 is 1, so 20% is 2. Now we can just increase one of the sides by 2, and decrease another side by 2. So our rectangle has dimensions of 12 x 8 instead of 10 x 10.

The original square had an area of 100, and the new rectangle has an area of 96. So the rectangle is 4 square units smaller, which is 4% smaller than the original square.

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Question

Side shown in the diagram of square below is equal to 21cm. What is the area of ?

Answer

To find the area of a quadrilateral, multiply length times width. In a square, since all sides are equal, is both the length and width.

$a^{2}=21\times21=441$

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Question

If Amy is carpeting her living room, which meaures feet by feet, how many square feet of carpet will she need?

Answer

To find the area of the floor, multiply the length of the room by the width (which is the same forumla used to find the area of a square). The equation can be written: $A=l\cdot w$

Substitute feet for and feet for :

$A=10\cdot 12$

Amy will need $120 ft^{2}$ of carpet.

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Question

A rectangle and a square have the same perimeter. The rectangle has length centimeters and width centimeters. Give the area of the square.

Answer

The perimeter of the rectangle is

$P = 2L + 2W = 2\cdot100 +2\cdot40 = 200 + 80 = 280$ centimeters.

This is also the perimeter of the square, so divide this by to get its sidelength:

$s = \frac{P}{4} = 280 \div 4 = 70$ centimeters.

The area is the square of this, or $A = s^{2} = 70^{2} = 4,900$ square centimeters.

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Question

Four squares have sidelengths 4 inches, 8 inches, 12 inches, and 16 inches. What is the average of their areas?

Answer

The areas of the four squares can be calculated by squaring their sidelengths. Add these areas, then divide by 4:

$\frac{4^{2}+8^{2}+12^{2}+16^{2}}{4} = \frac{16+64+144+256}{4} = \frac{480}{4} = 120$ square inches

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Question

Which of the following is equal to the area of a square with sidelength $1 \frac{1}{6}$ yards?

Answer

Multiply the sidelength by 36 to convert from yards to inches:

$1 \frac{1}{6} \times36 = \frac{7}{6} \times \frac{36}{1} = \frac{7}{1} \times \frac{6}{1} = 42\ inches$

Square this to get the area:

$42^{2} = 1,764$ square inches

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Question

What is the area of a square in which the length of one side is equal to $\sqrt{\frac{1}{4}+\frac{1}{3}}$ ?

Answer

The area of a square is equal to the product of one side multiplied by another side. Therefore, the area will be equal to:

$\sqrt{\frac{1}{4}+\frac{1}{3}}\cdot \sqrt{\frac{1}{4}+\frac{1}{3}}=\frac{1}{4}+\frac{1}{3}$

The next step is to convert the fractions being added together to a form in which they have a common denominator. This gives us:

$\frac{3}{12}+\frac{4}{12}=\frac{7}{12}$

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Question

One of the sides of a square on the coordinate plane has its endpoints at the points with coordinates and . What is the area of this square?

Answer

The length of a segment with endpoints and can be found using the distance formula with $x_{1} = 6$ , $y_{1} = 2$ , $x_{2} = -5$ , $y_{2} = 1$ :

$d = \sqrt{(x_{2}- x_{1})^{2}+(y_{2}-y_{1})^{2}}$

$= \sqrt{(6 - (-5))^{2}+(2-1)^{2}}$

$= \sqrt{11^{2}+1^{2}}$

$= \sqrt{121+1}$

$= \sqrt{122}$

This is the length of one side of the square, so the area is the square of this, or 122.

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Question

One of the sides of a square on the coordinate plane has its endpoint at the points with coordinates and , where and are both positive. Give the area of the square in terms of and .

Answer

The length of a segment with endpoints and can be found using the distance formula as follows:

$d = \sqrt{(x_{2}- x_{1})^{2}+(y_{2}-y_{1})^{2}}$

$d = \sqrt{((-B)-A)^{2}+(A-B)^{2}}$

$d = \sqrt{(-B-A)^{2}+(A-B)^{2}}$

$d = \sqrt{(-1)^{2}(B+A)^{2}+(A-B)^{2}}$

$d = \sqrt{1 \cdot (B^{2} +2AB +A^{2})+(A^{2}-2AB+B^{2})}$

$d = \sqrt{ (B^{2} +2AB +A^{2})+(A^{2}-2AB+B^{2})}$

$d = \sqrt{ 2A^{2} +2 B^{2}}$

This is the length of one side of the square, so the area is the square of this, or $2A^{2} +2 B^{2}$ .

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Question

One of the vertices of a square is at the origin. The square has area 13. Which of the following could be the vertex of the square opposite that at the origin?

Answer

Since a square is a rhombus, one way to calculate the area of a square is to take half the square of the length of a diagonal. If we let be the length of each diagonal, then

$\frac{1}{2} d^{2} = A$

$\frac{1}{2} d^{2} = 13$

$d^{2} =26$

$d = \sqrt{26}$

Therefore, we want to choose the point that is $\sqrt{26}$ units from the origin. Using the distance formula, we see that is such a point:

$d = \sqrt{(x_{2}- x_{1})^{2}+(y_{2}-y_{1})^{2}}$

$= \sqrt{(5-0)^{2}+(1-0)^{2}}$

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

$= \sqrt{25+1}$

$= \sqrt{26}$

Of the other points:

:

$d = \sqrt{(x_{2}- x_{1})^{2}+(y_{2}-y_{1})^{2}}$

$= \sqrt{(6-0)^{2}+(0-0)^{2}}$

$= \sqrt{6^{2}+0^{2}}$

$= \sqrt{36+0}$

$= \sqrt{36 }$

:

$d = \sqrt{(x_{2}- x_{1})^{2}+(y_{2}-y_{1})^{2}}$

$= \sqrt{(4-0)^{2}+(2-0)^{2}}$

$= \sqrt{4^{2}+2^{2}}$

$= \sqrt{16+4}$

$= \sqrt{20}$

:

$d = \sqrt{(x_{2}- x_{1})^{2}+(y_{2}-y_{1})^{2}}$

$= \sqrt{(3-0)^{2}+(3-0)^{2}}$

$= \sqrt{3^{2}+3^{2}}$

$= \sqrt{9+9}$

$= \sqrt{18}$

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Question

In the above diagram, the circle is inscribed inside the square. The circle has area 30. What is the area of the square?

Answer

In terms of , the area of the circle is equal to

$A_{1} = \pi r^{2}$ .

Each side of the square has length equal to the diameter, , so its area is the square of this, or

$A_{2} = (2r)^{2} = 4 r^{2}$

Therefore, the ratio of the area of the square to that of the circle is

$\frac{A_{2} }{A_{1} } = \frac{4 r^{2}}{\pi r^{2}}= \frac{4}{\pi }$

Therefore, the area of the circle is multiplied by this ratio to get the area of the square:

$A_{1} \cdot \frac{A_{2} }{A_{1} } =A_{1} \cdot \frac{4}{\pi }$

$A_{2} =A_{1} \cdot \frac{4}{\pi }$

Substituting:

$A_{2} = 30 \cdot \frac{4}{\pi } = \frac{120}{\pi }$

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Question

One of your holiday gifts is wrapped in a cube-shaped box.

If one of the edges has a length of 6 inches, what is the area of one side of the box?

Answer

One of your holiday gifts is wrapped in a cube-shaped box.

If one of the edges has a length of 6 inches, what is the area of one side of the box?

We are asked to find the area of one side of a cube, in other words, the area of a square.

We can find the area of a square by squaring the length of the side.

$A_{square}=s^2=(6in)^2=36in^2$

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Question

Find the area of a square with a base of 9cm.

Answer

To find the area of a square, we will use the following formula:

$\text{area of square} = l \cdot w$

where l is the length and w is the width of the square.

Now, we know the base (or length) of the square is 9cm. Because it is a square, all sides are equal. Therefore, the width is also 9cm.

Knowing this, we can substitute into the formula. We get

$\text{area of square} = 9\text{cm} \cdot 9\text{ccm}$

$\text{area of square} = 81\text{cm}^2$

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Question

While out walking, you find a strange, square-shaped piece of metal. If the side length of the piece is 26 inches, what is the area of the square?

Answer

While out walking, you find a strange, square-shaped piece of metal. If the side length of the piece is 26 inches, what is the area of the square?

To find the area of a square, simply square the side length:

So, our answer is:

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Question

What is the area of the square with a side length of $2\sqrt{3}$ ?

Answer

Write the formula for the area of a square.

Substitute the side into the formula.

$A=(2\sqrt{3})^2$

$A = 4\cdot 3 = 12$

The answer is:

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Question

What is the diagonal of a square with a side of 4?

Answer

Squares have all congruent sides. To find the diagonal, first recognize that you're dealing with an isoceles triangle when you draw the diagonal in the square. That means that two of the sides are congruent in the triangle. Thus, it's a special 45-45-90 triangle. In such triangles, the sides are x and the hypotenuse is $x\sqrt{2}$ . Since we know x is 4, we can plug in 4 to the expression $x\sqrt{2}$ . Thus, the answer is $4\sqrt{2}$ .

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Question

You recently bought some special filter paper for a laboratory apparatus. The paper comes in square sheets, but you want to cut it into two equal triangle-shaped pieces. If the square sheets have a side length of , what will the length of the hypotenuse of the triangles be?

Answer

You recently bought some special filter paper for a laboratory apparatus. The paper comes in square sheets, but you want to cut it into two equal triangle-shaped pieces. If the square sheets have a side length of , what will the length of the hypotenuse of the triangles be?

This problem is trying to distract you by thinking of triangles. What we are really asked to find here is the length of the diagonal of a square with sides of 15 inches.

Splitting a square along its diagonal yields two 45/45/90 triangles. If you know the ratios for 45/45/90 triangles, you can find the answer very quickly.

Think:

$45/45/90\rightarrow x/x/x\sqrt{2}$

Meaning that if the two short sides are x units long, the hypotenuse will be x times the square root of two units long.

In our current case, our short sides are 15 inches long, so our hypotenuse will be

$15\sqrt{2} in$

You could also solve this with Pythagorean Theorem.

a and b are both 15 in, so we can solve.

$c=\sqrt{225in^2+225in^2}=\sqrt{450in^2}=\sqrt{15^2in^2*2}=15\sqrt{2}in$

So,our answer is

$15\sqrt{2} in$

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Question

While out walking, you find a strange, square-shaped piece of metal. If the side length of the piece is 26 inches, what is the length of the diagonal?

Answer

While out walking, you find a strange, square-shaped piece of metal. If the side length of the piece is 26 inches, what is the length of the diagonal?

To find the diagonal of a square, we can recognize one of two things.

The diagonal of a square creates a right triangle, and we can use Pythagorean theorem to find our diagonal.

The diagonal of a square creates two 45/45/90 triangles, with side length ratios of $x/x/\sqrt{2}x$

Using 2), we can find that the diagonal of the square must be $26\sqrt{2} in$

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Question

Your new friend has a very small, square-shaped dorm room. She tells you that it is only 225 square feet. Assuming this is true, what is the diagonal distance from one corner of her room to the other?

Answer

Your new friend has a very small, square-shaped dorm room. She tells you that it is only 225 square feet. Assuming this is true, what is the diagonal distance from one corner of her room to the other?

So, we need to find the diagonal of a square. First, we need to find the side length.

Let's begin with our formula for the area of a square:

where s is our side length and A is our area.

With this formula, we can solve for our side length by plugging in our area and square rooting both sides.

$s=\sqrt{225ft^2}=15ft$

Now, to find the diagonal, we can think of an isosceles right triangle, where the two equal sides are 15 ft. This is also a 45/45/90 triangle, which means the side lengths follow the ratio of $x, x , x \sqrt{2}$ .

This means our answer is $15 \sqrt{2}ft$ .

We could also find this by using Pythagorean Theorem.

$c^2=\sqrt{450ft^2}=\sqrt{15^2*2}ft=15\sqrt{2}ft$

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Question

The area of a square is 169 square inches. What is the length of one side ( in the diagram below)?

Answer

Area of a quadrilateral is found by length times width. In a square, these are the same, so the length of side is a number that, when multiplied by itself is equal to 169.

In other words, take the square root of 169 to find the length of .

$\sqrt{169}=13$

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