DSQ: Calculating the area of a rectangle - GMAT Quantitative Reasoning
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Data sufficiency question- do not actually solve the question
Does the square or rectangle have a greater area?
1. The perimeter of both the square and rectangle are equal.
2. The rectangle does not have four equal sides.
Data sufficiency question- do not actually solve the question
Does the square or rectangle have a greater area?
1. The perimeter of both the square and rectangle are equal.
2. The rectangle does not have four equal sides.
When a square and rectangle have the same perimeter, the square will have a larger area because having 4 equal sides maximizes the area. However, from statement 1, it is impossible to tell if the rectangle is also a square. When the information from statement 2 is combined, we can conclude that the rectangle is not also a square.
When a square and rectangle have the same perimeter, the square will have a larger area because having 4 equal sides maximizes the area. However, from statement 1, it is impossible to tell if the rectangle is also a square. When the information from statement 2 is combined, we can conclude that the rectangle is not also a square.
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What is the area of a rectangle?
Statement 1: The length of its diagonal is 25.
Statement 2: The diagonal and either of its longer sides form a 
angle.
What is the area of a rectangle?
Statement 1: The length of its diagonal is 25.
Statement 2: The diagonal and either of its longer sides form a
To find the rectangle, you need the length and the width.
If you know the diagonal 
and the angle 
it forms with one of the longer sides, you can use trigonometry to find both length and width:
From there, the area follows.
If you know only the diagonal, you have insufficient information; the length and width can vary according to that angle. If you only know the angle, you can discern the proportions of the sides, but not the actual lengths.
To find the rectangle, you need the length and the width.
If you know the diagonal
From there, the area follows.
If you know only the diagonal, you have insufficient information; the length and width can vary according to that angle. If you only know the angle, you can discern the proportions of the sides, but not the actual lengths.
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A rectangle has vertices 
,
where 
Of the four quadrants, which one includes the greatest portion of the rectangle?
Statement 1: 
Statement 2: 
A rectangle has vertices
where
Of the four quadrants, which one includes the greatest portion of the rectangle?
Statement 1:
Statement 2:
The portion of the rectangle to the right of the 
-axis has area 
; to the left, 
. From Statement 1 alone, since 
, 
, and the portion of the rectangle on the right is greater than the portion on the left. However, this is all we can determine.
By a similar argument, from Statement 2 alone, the portion of the rectangle above is greater than the portion below, but this is all we can determine.
From both statements together, however, we can compare the portions of the rectangles in the four quadrants. The areas of each are:
Quadrant I: 
Quadrant II: 
Quadrant III: 
Quadrant IV: 
Since 
and 
, 
is the greatest of the four quantities, and we see that Quadrant I includes the lion's share of the rectangle.
The portion of the rectangle to the right of the
By a similar argument, from Statement 2 alone, the portion of the rectangle above is greater than the portion below, but this is all we can determine.
From both statements together, however, we can compare the portions of the rectangles in the four quadrants. The areas of each are:
Quadrant I:
Quadrant II:
Quadrant III:
Quadrant IV:
Since
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Rectangle 
is inscribed in a circle. What is its area?
Statement 1; The circle has area 
.
Statement 2: 
Rectangle
Statement 1; The circle has area
Statement 2:
The figure referenced is below:
Assume Statement 1 alone. 
is a diagonal of the rectangle, and also a diameter of the circle. Statement 1 gives the area of the circle, from which the radius, and, subsequently, 
, can be calculated. However, infinitely many rectangles of different areas can be constructed in this circle, so without any further information, it is not clear what the sidelengths are - and what the area is.
Assume Statement 2 alone. This statement only gives the length of one side. Without any further information, the area of the rectangle is unknown.
Now assume both statements are true. 
can be calculated, and 
is given, so the Pythagorean Theorem can be used to find 
. The area of the rectangle is the product 
, so the two statements together are sufficient.
The figure referenced is below:
Assume Statement 1 alone.
Assume Statement 2 alone. This statement only gives the length of one side. Without any further information, the area of the rectangle is unknown.
Now assume both statements are true.
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Rectangle 
is inscribed in a circle. What is the area of the rectangle?
Statement 1: The circle has area 77.
Statement 2: The rectangle is a square.
Rectangle
Statement 1: The circle has area 77.
Statement 2: The rectangle is a square.
The figure referenced is below (note that the figure itself assumes Statement 2, but this is not known from Statement 1):
Assume Statement 1 alone. A diagonal of a rectangle inscribed inside a circle is a diameter of the circle. Statement 1 gives the area of the circle, from which the radius, and, subsequently, 
, can be calculated. However, infinitely many rectangles of different areas can be constructed in a given circle, so without any further information, it is not clear what the sidelengths are - and what the area is.
Assume Statement 2 alone. It follows that all of the sides of the rectangle/square are congruent, but without the common sidelength, the area of the square cannot be calculated.
Assume both statements. 
can be calculated, and the area of the square can be calculated to be 
.
The figure referenced is below (note that the figure itself assumes Statement 2, but this is not known from Statement 1):
Assume Statement 1 alone. A diagonal of a rectangle inscribed inside a circle is a diameter of the circle. Statement 1 gives the area of the circle, from which the radius, and, subsequently,
Assume Statement 2 alone. It follows that all of the sides of the rectangle/square are congruent, but without the common sidelength, the area of the square cannot be calculated.
Assume both statements.
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Give the area of a given rectangle.
Statement 1: The perimeter of the rectangle is 10.
Statement 2: All sides of the rectangle have a length equal to a prime integer.
Give the area of a given rectangle.
Statement 1: The perimeter of the rectangle is 10.
Statement 2: All sides of the rectangle have a length equal to a prime integer.
A a rectangle with sides of length 1 and 4 and a rectangle with sides of length 2 and 3 both have perimeter 10, but they have different areas ( 4 and 6, respectively), making Statement 1 alone inconclusive. Statement 2 is inconclusive, there being infinitely many primes.
Assume both statements.
Then
Since 
and 
are both prime integers, one must be 2 and the other must be 3 (1 and 4 cannot be a possibility, since 1 is not a prime). It does not matter which is which, so the numbers can be multiplied to obtain area 6.
A a rectangle with sides of length 1 and 4 and a rectangle with sides of length 2 and 3 both have perimeter 10, but they have different areas ( 4 and 6, respectively), making Statement 1 alone inconclusive. Statement 2 is inconclusive, there being infinitely many primes.
Assume both statements.
Then
Since
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Give the area of a given rectangle.
Statement 1: The perimeter of the rectangle is 36.
Statement 2: All sides of the rectangle have a length equal to an odd prime integer.
Give the area of a given rectangle.
Statement 1: The perimeter of the rectangle is 36.
Statement 2: All sides of the rectangle have a length equal to an odd prime integer.
Assume both statements. Then from Statement 1, it follows that:
There are two pairs of odd primes that add up to 18 - (5,13), in which case the area is 65, and (7,11), in which case the area is 77. The two statements together are inconclusive.
Assume both statements. Then from Statement 1, it follows that:
There are two pairs of odd primes that add up to 18 - (5,13), in which case the area is 65, and (7,11), in which case the area is 77. The two statements together are inconclusive.
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