Rectangles - ISEE Upper Level Quantitative Reasoning

The sidelength of a square with area square centimeters is centimeters; its perimeter, as well as that of the rectangle, is therefore centimeters.

Using the formula for the perimeter of a rectangle, substitute and solve for as follows:

The total area of the rectangle is

.

The area of the portion of the rectangle in Quadrant I is

.

Therefore, the portion of the rectangle in Quadrant I is

.

In a rectangle in which the width is x while the length is 4x, the first step is to solve for x. If , the value of x can be found by dividing each side of this equation by 3.

Doing so gives us the information that x is equal to 3.

Thus, the area is equal to:

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

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

Now, we know the width of the rectangle is 5in. We also know the length is three times the width. Therefore, the length is 15in.

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

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

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

Now, we know the width of the rectangle is 7cm. We also know the length is four times the width. Therefore, the length is 28cm.

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

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

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

Now, we know the width of the rectangle is 8cm. We also know the length of the rectangle is four times the width. Therefore, the length of the rectangle is 32cm.

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

You are designing a poster to put on the front of your refrigerator. If the refrigerator door is 2 feet wide by 4.5 feet tall, what is the area of the largest poster you could fit on the door?

We need to find the area of a shape. Given the context of a refrigerator door and a poster, we can assume that the poster will be a rectangle. To find the area of a rectangle, we need to multiply length and width.

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

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

Now, we know the length of the rectangle is 12in. We also know the width of the rectangle is a third of the length. Therefore, the width is 4in.

So, we get

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

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

Now, we know the width of the rectangle is 9cm. We also know the length of the rectangle is three times the width. Therefore, the length is 27cm. So, we can substitute.

Without more information we cannot determine the width of a rectangle based on its area alone.

If we know this is a square and therefore that all sides are equal, we could simply take the square root of 100, which is 10.

However, we do not know that all sides are equal. We need to know the length of the rectangle to solve this problem. Therefore the best we can do is see what the equation would look like if we knew the length.

or

where is the width and is the length.

Note: the incorrect answer choices are all factors of 100 and possible answers, but we can't be sure which, if any, are the right one.

Substitute in the formula for the perimeter of a rectangle:

Let be the length. The area of the rectangle is . Replace and solve for :

Let be the length. The perimeter of the rectangle is .

Replace and solve for :

You have a shed that you use for storing your gardening supplies and outdoor tools. If the shed has a footprint of and a width of 16 ft, what is the length of the shed? (assume the base is a rectangle)

We are given a rectangle with a known area and width, and asked to find the length.

To do so, recall this formula:

We know A and w, so simply work backwards to solve for l

So, our answer is 17.75 ft

Your geometry book has a rectangular front cover which is 12 inches by 8 inches.

What is the area of your book cover?

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

Plug in our knowns and solve:

The sidelength of a square with area square centimeters is centimeters; its perimeter, as well as that of the rectangle, is therefore centimeters.

Using the formula for the perimeter of a rectangle, substitute and solve for as follows:

The total area of the rectangle is

.

The area of the portion of the rectangle in Quadrant I is

.

Therefore, the portion of the rectangle in Quadrant I is

.

In a rectangle in which the width is x while the length is 4x, the first step is to solve for x. If , the value of x can be found by dividing each side of this equation by 3.

Doing so gives us the information that x is equal to 3.

Thus, the area is equal to:

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

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

Now, we know the width of the rectangle is 5in. We also know the length is three times the width. Therefore, the length is 15in.

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