ISEE Upper Level Math : Quadrilaterals

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

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Example Questions

Example Question #11 : Quadrilaterals

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 perimeter of the square?

Possible Answers:

\displaystyle 104 in

\displaystyle 52 in

\displaystyle 78 in

\displaystyle 676in

Correct answer:

\displaystyle 104 in

Explanation:

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 perimeter of the square?

To find the perimeter of a square, simply multiply the side length by 4. We can do this because perimeter is the distance around the outside of a shape, and squares have 4 equal sides.

\displaystyle P=4s=(26in)4=104in

 

Example Question #231 : Isee Upper Level (Grades 9 12) Mathematics Achievement

Find the perimeter of a square with a width of 18in.

Possible Answers:

\displaystyle 72\text{in}

\displaystyle 64\text{in}

\displaystyle 36\text{in}

\displaystyle 96\text{in}

\displaystyle \text{There is not enough information to solve the problem.}

Correct answer:

\displaystyle 72\text{in}

Explanation:

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

\displaystyle P = a+b+c+d

where a, b, c, and d are the lengths of the sides of the square.

Now, we know the width of the square is 18in. Because it is a square, all sides are equal. Therefore, all sides are 18in.

So, we get

\displaystyle P = 18\text{in} + 18\text{in} + 18\text{in} + 18\text{in}

\displaystyle P = 72\text{in}

Example Question #8 : How To Find The Perimeter Of A Square

Find the perimeter of a square with a side length of \displaystyle 3x^2.

Possible Answers:

\displaystyle 9x^2

\displaystyle 12x^2

\displaystyle 3x^6

\displaystyle 7x^2

\displaystyle 3x^8

Correct answer:

\displaystyle 12x^2

Explanation:

A square has 4 equal sides.

Write the formula for the perimeter of a square.

\displaystyle P=4s

Substitute the side length.

\displaystyle P=4(3x^2)

Evaluate the terms on the right.

The answer is:  \displaystyle 12x^2

Example Question #9 : How To Find The Perimeter Of A Square

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 perimeter of her room?

 

Possible Answers:

\displaystyle 15ft

\displaystyle 120ft

\displaystyle 60 ft

Not enough information to solve the problem

Correct answer:

\displaystyle 60 ft

Explanation:

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 perimeter of her room?

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

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

\displaystyle A=s^2

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.

\displaystyle 225 ft^2=s^2

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

Now, we are close but not quite done. We need to multiply our side length by 4, because a square always has 4 equal sides.

\displaystyle P=4s=4(15ft)=60ft

Example Question #11 : Quadrilaterals

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?

Possible Answers:

the area remains the same

decreased by 10%

decreased by 4%

increased by 20%

Correct answer:

decreased by 4%

Explanation:

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.

Example Question #12 : Quadrilaterals

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

342px-square_-_geometry.svg

Possible Answers:

\displaystyle 441\ cm^{2}

\displaystyle 84\ cm^{2}

Cannot be determined

\displaystyle 210\ cm^{2}

\displaystyle 42\ cm^{2}

Correct answer:

\displaystyle 441\ cm^{2}

Explanation:

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

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

Example Question #13 : Quadrilaterals

If Amy is carpeting her living room, which meaures \displaystyle 10 feet by \displaystyle 12 feet, how many square feet of carpet will she need?

Possible Answers:

\displaystyle 44 ft^{2}

\displaystyle 144 ft^{2}

\displaystyle 120 ft^{2}

\displaystyle 100 ft^{2}

\displaystyle 22 ft^{2}

Correct answer:

\displaystyle 120 ft^{2}

Explanation:

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: \displaystyle A=l\cdot w

Substitute \displaystyle 10 feet for \displaystyle l and \displaystyle 12 feet for \displaystyle w:

\displaystyle A=10\cdot 12

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

Example Question #244 : Isee Upper Level (Grades 9 12) Mathematics Achievement

A rectangle and a square have the same perimeter. The rectangle has length \displaystyle 100 centimeters and width \displaystyle 40 centimeters. Give the area of the square.

Possible Answers:

\displaystyle 8,000 \textrm{ cm}^{2}

\displaystyle 3,600 \textrm{ cm}^{2}

\displaystyle 19,600 \textrm{ cm}^{2}

\displaystyle 4,900 \textrm{ cm}^{2}

\displaystyle 4,000 \textrm{ cm}^{2}

Correct answer:

\displaystyle 4,900 \textrm{ cm}^{2}

Explanation:

The perimeter of the rectangle is

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

This is also the perimeter of the square, so divide this by \displaystyle 4 to get its sidelength:

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

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

Example Question #1 : How To Find The Area Of A Square

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

Possible Answers:

\displaystyle 144\textrm{ in} ^{2}

\displaystyle 200\textrm{ in} ^{2}

\displaystyle 120\textrm{ in} ^{2}

\displaystyle 64\textrm{ in} ^{2}

\displaystyle 100\textrm{ in} ^{2}

Correct answer:

\displaystyle 120\textrm{ in} ^{2}

Explanation:

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

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

Example Question #2 : How To Find The Area Of A Square

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

Possible Answers:

\displaystyle 2,304 \textrm{ in}^{2}

\displaystyle 1,764 \textrm{ in}^{2}

\displaystyle 1,225 \textrm{ in}^{2}

\displaystyle 2,025 \textrm{ in}^{2}

Correct answer:

\displaystyle 1,764 \textrm{ in}^{2}

Explanation:

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

\displaystyle 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:

\displaystyle 42^{2} = 1,764 square inches

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