Rectangles - GMAT Quantitative
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The area of a square that has sides with a length of 12 inches is equal to the area of a rectangle. If the rectangle has a width of 3 inches, what is the length of the rectangle?
The area of a square that has sides with a length of 12 inches is equal to the area of a rectangle. If the rectangle has a width of 3 inches, what is the length of the rectangle?
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If the area of the rectangle is equal to the area of the square, then it must have an area of $144in^{2}$ (12times 12). If the rectangle has an area of 144 $in^{2}$ and a side with a lenth of 3 inches, then the equation to solve the problem would be 144=3x, where x is the length of the rectangle. The solution:
$\frac{144}{3}$ = 48.
If the area of the rectangle is equal to the area of the square, then it must have an area of $144in^{2}$ (12times 12). If the rectangle has an area of 144 $in^{2}$ and a side with a lenth of 3 inches, then the equation to solve the problem would be 144=3x, where x is the length of the rectangle. The solution:
$\frac{144}{3}$ = 48.
Note: figure NOT drawn to scale
Give the area of the above rectangle.
Note: figure NOT drawn to scale
Give the area of the above rectangle.
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The area of a rectangle is the product of its length and width;
The area of a rectangle is the product of its length and width;
A rectangle twice as long as it is wide has perimeter 
. Write its area in terms of 
.
A rectangle twice as long as it is wide has perimeter
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Let 
be the width of the rectangle; then its length is 
, and its perimeter is
Set this equal to 
and solve for 
:
The width is 
and the length is 
, so multiply these expressions to get the area:
Let
Set this equal to
The width is
A rectangle has its vertices at 
. What part, in percent, of the rectangle is located in Quadrant III?
A rectangle has its vertices at
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A rectangle with vertices 
has width 
and height 
, thereby having area 
.
The portion of the rectangle in Quadrant III is a rectangle with vertices
.
It has width 
and height 
, thereby having area 
.
Therefore, 
of the rectangle is in Quadrant III; this is equal to
A rectangle with vertices
The portion of the rectangle in Quadrant III is a rectangle with vertices
It has width
Therefore,
A rectangle has its vertices at 
. What percentage of the rectangle is located in Quadrant IV?
A rectangle has its vertices at
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A rectangle with vertices has width and height ; it follows that its area is .
The portion of the rectangle in Quadrant IV has vertices 
. Its width is 
, and its height is 
, so its area is 
.
Therefore, 
, or 
, of this rectangle is in Quadrant IV.
A rectangle with vertices
The portion of the rectangle in Quadrant IV has vertices
Therefore,
What is the area of a rectangle given the length of 
and width of 
?
What is the area of a rectangle given the length of
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To find the area of a rectangle, you must use the following formula:
To find the area of a rectangle, you must use the following formula:
What polynomial represents the area of a rectangle with length 
and width 
?
What polynomial represents the area of a rectangle with length
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The area of a rectangle is the product of the length and the width. The expression 
can be multplied by noting that this is the product of the sum and the difference of the same two terms; its product is the difference of the squares of the terms, or
The area of a rectangle is the product of the length and the width. The expression
The perimeter of a rectangle is 
and its length is 
times the width. What is the area?
The perimeter of a rectangle is
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The perimeter of a rectangle is the sum of all four sides, that is: 
since 
, we can rewrite the equation as:
We are being asked for the area so we still aren't done. The area of a rectangle is the product of the width and length. We know what the width is so we can find the length and then take their product.
The perimeter of a rectangle is the sum of all four sides, that is:
since
We are being asked for the area so we still aren't done. The area of a rectangle is the product of the width and length. We know what the width is so we can find the length and then take their product.
Find the area of a rectangle whose side lengths are 
.
Find the area of a rectangle whose side lengths are
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To calculate area, multiply width times height. Thus,
To calculate area, multiply width times height. Thus,
Mark is building a garden with raised beds. One side of the garden will be 10 feet long and the other will be 5 less than three times the first side. What area of will Mark's garden be?
Mark is building a garden with raised beds. One side of the garden will be 10 feet long and the other will be 5 less than three times the first side. What area of will Mark's garden be?
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Mark is building a garden with raised beds. One side of the garden will be 10 feet long and the other will be 5 less than three times the first side. What area of will Mark's garden be?
This problem asks us to find the area of a rectangle. We are given one side and asked to find the other. To find the other, we need to use the provided clues.
"...five less..." 
"...three times the first side..." 
or 
So put it together:
Next, find the area via the following formula:
So our answer is:
Mark is building a garden with raised beds. One side of the garden will be 10 feet long and the other will be 5 less than three times the first side. What area of will Mark's garden be?
This problem asks us to find the area of a rectangle. We are given one side and asked to find the other. To find the other, we need to use the provided clues.
"...five less..."
"...three times the first side..."
So put it together:
Next, find the area via the following formula:
So our answer is:
Find the area of a rectangle whose width is 
and length is 
.
Find the area of a rectangle whose width is
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To find area, simply multiply length times width. Thus
To find area, simply multiply length times width. Thus
In the above diagram,
.
and 
. Give the area of 
.
In the above diagram,
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, so
The area of the rectangle is
The area of the rectangle is
, and 
.
All the following quantities MUST be equal to 2 except for .
All the following quantities MUST be equal to 2 except for .
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The two rectangles are similar with similarity ratio 2.
Corresponding sides of similar rectangles are in proportion, so
.
Since opposite sides of a rectangle are congruent, 
, so
.
and 
are diagonals of the rectangle. If they are constructed, then, since 
and 
(both are right angles), by the Side-Angle-Side Similarity Theorem, 
. By similarity, 
.
The ratio of the perimeters of the rectangles is
,
It follows from 
and a property of proportions that this ratio is equal to 
.
However,
is not a ratio of corresponding sides of the rectangle, so it does not have any restrictions on it. This is the correct choice.
The two rectangles are similar with similarity ratio 2.
Corresponding sides of similar rectangles are in proportion, so
Since opposite sides of a rectangle are congruent,
The ratio of the perimeters of the rectangles is
It follows from
However,
is not a ratio of corresponding sides of the rectangle, so it does not have any restrictions on it. This is the correct choice.
;
.
Which of the following must be equal to 
?
Which of the following must be equal to
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, so 
; this is the similarity ratio of 
to 
.
and 
are not corresponding sides, nor are they congruent to corresponding sides, so it may or may not be true that 
.
The ratio of the perimeters of similar rectangles is the same as their actual similarity ratio, but the choice gives the rectangles in the original order; the quotient of the perimeters as given is 3, not 
.
The ratio of the areas of similar rectangles is the square of the similarity ratio, so the quotient in the given choice is 
.
However, by similarity, and the fact that opposite sides 
and 
are congruent,
.
The correct choice is 
.
The ratio of the perimeters of similar rectangles is the same as their actual similarity ratio, but the choice gives the rectangles in the original order; the quotient of the perimeters as given is 3, not
The ratio of the areas of similar rectangles is the square of the similarity ratio, so the quotient in the given choice is
However, by similarity, and the fact that opposite sides
The correct choice is
Note: Figure NOT drawn to scale.
Refer to the above figure. 
.
.
Give the ratio of the area of the shaded region to the area of 
.
Note: Figure NOT drawn to scale.
Refer to the above figure.
Give the ratio of the area of the shaded region to the area of
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The ratio of the areas of similar rectangles is the square of the similarity ratio, so the ratio of the area of 
to that of 
is
.
So if the area of 
is 
, the area of 
is 
The area of the shaded region is the difference between the areas of the rectangles, making this area
.
The desired ratio is 24 to 25.
The ratio of the areas of similar rectangles is the square of the similarity ratio, so the ratio of the area of
So if the area of
The area of the shaded region is the difference between the areas of the rectangles, making this area
The desired ratio is 24 to 25.
A certain rectangle has a length of 
and a width of 
. Which of the following dimensions would another rectangle need in order for it to be similar?
A certain rectangle has a length of
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In order for two rectangles to be similar, the ratio of their dimensions must be equal. We can first check the ratio of the length to the width for the given rectangle, and then see which option has the same ratio, which will tell us whether or not the rectangle is similar:
So in order for a rectangle to be similar, the ratio of its length to its width must be the same. All we must do then is check the answer options, in no particular order, for the rectangle with the same ratio:
A rectangle with a length of 
and a width of 
has the same ratio of dimensions as a rectangle with a length of 
and a width of 
, so these two rectangles are similar.
In order for two rectangles to be similar, the ratio of their dimensions must be equal. We can first check the ratio of the length to the width for the given rectangle, and then see which option has the same ratio, which will tell us whether or not the rectangle is similar:
So in order for a rectangle to be similar, the ratio of its length to its width must be the same. All we must do then is check the answer options, in no particular order, for the rectangle with the same ratio:
A rectangle with a length of
An engineer is making a scale model of a building. The real building needs to have a width of 
and a length of 
. If the engineer's scale model has a width of 
, what does the length of the model need to be?
An engineer is making a scale model of a building. The real building needs to have a width of
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An engineer is making a scale model of a building. The real building needs to have a width of 
and a length of 
. If the engineer's scale model has a width of 
, what does the length of the model need to be?
To begin, we need to know what a scale model is. A scale model is a smaller version of something that is "to scale." In other words, it is similar but not congruent.
So, we find a length that will make the model accurate, we need a ratio. Try the following:
An engineer is making a scale model of a building. The real building needs to have a width of
To begin, we need to know what a scale model is. A scale model is a smaller version of something that is "to scale." In other words, it is similar but not congruent.
So, we find a length that will make the model accurate, we need a ratio. Try the following:
A rectangle has a length of 
and width of 
. Solve for the perimeter.
A rectangle has a length of
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A farmer decides to build a fence two feet around his rectangular field. The field is 
feet long and 
feet wide. How long should the fencing be in order to build the fence around the field?
A farmer decides to build a fence two feet around his rectangular field. The field is
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The fence is 2 feet around the field. Therefore the area actually enclosed by the fence is 4 feet longer and 4 feet wider than the field.
The length of fencing needed is the perimeter of the area enclosed which is calculated as follows:
The fencing should be 90 feet.
The fence is 2 feet around the field. Therefore the area actually enclosed by the fence is 4 feet longer and 4 feet wider than the field.
The length of fencing needed is the perimeter of the area enclosed which is calculated as follows:
The fencing should be 90 feet.
If a rectangle has an area of 
and a length of 
, what is its perimeter?
If a rectangle has an area of
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To find the perimeter, we need the length and the width. We are only given the length, so first we must find the width using the given area:
The perimeter is simply two times the length plus two times the width, so we can now use the known length and width to calculate the perimeter:
To find the perimeter, we need the length and the width. We are only given the length, so first we must find the width using the given area:
The perimeter is simply two times the length plus two times the width, so we can now use the known length and width to calculate the perimeter: