Squares

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ISEE Upper Level Quantitative Reasoning › Squares

Questions 1 - 10

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 length of one side of her room?

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 length of one side of her room?

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.

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

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 length of one side of her room?

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 length of one side of her room?

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.

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

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?

decreased by 4%

increased by 20%

the area remains the same

decreased by 10%

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.

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?

decreased by 4%

increased by 20%

the area remains the same

decreased by 10%

Explanation

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

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.

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

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

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

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

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

Explanation

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

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

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

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

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

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

Explanation

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

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

$81\text{cm}^2$

$18\text{cm}^2$

$36\text{cm}^2$

$54\text{cm}^2$

$72\text{cm}^2$

Explanation

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

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

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

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?

$15\sqrt{2}ft$

$15\sqrt{3}ft$

Explanation

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.

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

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 .