Quadrilaterals - GRE Quantitative Reasoning
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In a given parallelogram, the measure of one of the interior angles is 25 degrees less than another. What is the approximate measure rounded to the nearest degree of the larger angle?
In a given parallelogram, the measure of one of the interior angles is 25 degrees less than another. What is the approximate measure rounded to the nearest degree of the larger angle?
There are two components to solving this geometry puzzle. First, one must be aware that the sum of the measures of the interior angles of a parallelogram is 360 degrees (sum of the interior angles of a figure = 180(n-2), where n is the number of sides of the figure). Second, one must know that the other two interior angles are doubles of those given here. Thus if we assign one interior angle as x and the other as x-25, we find that x + x + (x-25) + (x-25) = 360. Combining like terms leads to the equation 4x-50=360. Solving for x we find that x = 410/4, 102.5, or approximately 103 degrees. Since x is the measure of the larger angle, this is our answer.
There are two components to solving this geometry puzzle. First, one must be aware that the sum of the measures of the interior angles of a parallelogram is 360 degrees (sum of the interior angles of a figure = 180(n-2), where n is the number of sides of the figure). Second, one must know that the other two interior angles are doubles of those given here. Thus if we assign one interior angle as x and the other as x-25, we find that x + x + (x-25) + (x-25) = 360. Combining like terms leads to the equation 4x-50=360. Solving for x we find that x = 410/4, 102.5, or approximately 103 degrees. Since x is the measure of the larger angle, this is our answer.
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Figure 
is a parallelogram.
What is 
in the figure above?
Figure
What is
Because of the character of parallelograms, we know that our figure can be redrawn as follows:
Because it is a four-sided figure, we know that the sum of the angles must be 
. Thus, we know:
Solving for 
, we get:
Because of the character of parallelograms, we know that our figure can be redrawn as follows:
Because it is a four-sided figure, we know that the sum of the angles must be
Solving for
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Figure 
is a parallelogram.
Quantity A: The largest angle of 
.
Quantity B: 
Which of the following is true?
Figure
Quantity A: The largest angle of
Quantity B:
Which of the following is true?
By using the properties of parallelograms along with those of supplementary angles, we can rewrite our figure as follows:
Recall, for example, that angle 
is equal to:
, hence 
Now, you know that these angles can all be added up to 
. You should also know that 
Therefore, you can write:
Simplifying, you get:
Now, this means that:
and 
. Thus, the two values are equal.
By using the properties of parallelograms along with those of supplementary angles, we can rewrite our figure as follows:
Recall, for example, that angle
Now, you know that these angles can all be added up to
Therefore, you can write:
Simplifying, you get:
Now, this means that:
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The perimeter of a rectangle is 14, and the diagonal connecting two vertices is 5.
Quantity A: 13
Quantity B: The area of the rectangle
The perimeter of a rectangle is 14, and the diagonal connecting two vertices is 5.
Quantity A: 13
Quantity B: The area of the rectangle
One potentially helpful first step is to draw the rectangle described in the problem statement:
After that, it's a matter of using the other information given. The perimeter is given as 14, and can be written in terms of the length and width of the rectangle:
Furthermore, notice that the diagonal forms the hypotenuse of a right triangle. The Pythagorean Theorem may be applied:
This provides two equations and two unknowns. Redefining the first equation to isolate 
gives:
Plugging this into the second equation in turn gives:
Which can be reduced to:
or
Note that there are two possibile values for 
; 3 or 4. The one chosen is irrelevant. Choosing a value 3, it is possible to then find a value for 
:
This in turn allows for the definition of the rectangle's area:
So Quantity B is 12, which is less than Quantity A.
One potentially helpful first step is to draw the rectangle described in the problem statement:
After that, it's a matter of using the other information given. The perimeter is given as 14, and can be written in terms of the length and width of the rectangle:
Furthermore, notice that the diagonal forms the hypotenuse of a right triangle. The Pythagorean Theorem may be applied:
This provides two equations and two unknowns. Redefining the first equation to isolate
Plugging this into the second equation in turn gives:
Which can be reduced to:
or
Note that there are two possibile values for
This in turn allows for the definition of the rectangle's area:
So Quantity B is 12, which is less than Quantity A.
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One rectangle has a height of 
and a width of 
. Which of the following is a possible perimeter of a similar rectangle, having one side that is 
?
One rectangle has a height of
Based on the information given, we know that 
could be either the longer or the shorter side of the similar rectangle. Similar rectangles have proportional sides. We might need to test both, but let us begin with the easier proportion, namely:
as 
For this proportion, you really do not even need fractions. You know that 
must be 
.
This means that the figure would have a perimeter of 
Luckily, this is one of the answers!
Based on the information given, we know that
For this proportion, you really do not even need fractions. You know that
This means that the figure would have a perimeter of
Luckily, this is one of the answers!
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One rectangle has sides of 
and 
. Which of the following pairs could be the sides of a rectangle similar to this one?
One rectangle has sides of
For this problem, you need to find the pair of sides that would reduce to the same ratio as the original set of sides. This is a little tricky at first, but consider the set:
and 
For this, you have:
Now, if you factor out 
, you have:
Thus, the proportions are the same, meaning that the two rectangles would be similar.
For this problem, you need to find the pair of sides that would reduce to the same ratio as the original set of sides. This is a little tricky at first, but consider the set:
For this, you have:
Now, if you factor out
Thus, the proportions are the same, meaning that the two rectangles would be similar.
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A parallelogram has a base of 
and a height of 
. Find the area of the parallelogram.
A parallelogram has a base of
By definition a parallelogram has two sets of opposite sides that are congruent/parallel. However to find the area of a paralleogram, you need to know the base and height lengths. Since this problem provides both the base and height measurements, apply the formula:
The solution is:
By definition a parallelogram has two sets of opposite sides that are congruent/parallel. However to find the area of a paralleogram, you need to know the base and height lengths. Since this problem provides both the base and height measurements, apply the formula:
The solution is:
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A parallelogram has a base of 
and a height measurement that is 
the base length. Find the area of the parallelogram.
A parallelogram has a base of
By definition a parallelogram has two sets of opposite sides that are congruent/parallel. However, to find the area of a paralleogram, you need to know the base and height lengths. Since this problem provides both the base and height measurements, apply the formula:
Before applying the formula you must find 
of 
.
The final solution is:
By definition a parallelogram has two sets of opposite sides that are congruent/parallel. However, to find the area of a paralleogram, you need to know the base and height lengths. Since this problem provides both the base and height measurements, apply the formula:
Before applying the formula you must find
The final solution is:
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Using the parallelogram shown above, find the area.
Using the parallelogram shown above, find the area.
This problem provides both the base and height measurements, thus apply the formula:
To find an equivalent answer in inches, you must convert the measurements to inches FIRST, and then multiply:
Therefore, our area in square inches is:
This problem provides both the base and height measurements, thus apply the formula:
To find an equivalent answer in inches, you must convert the measurements to inches FIRST, and then multiply:
Therefore, our area in square inches is:
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A parallelogram has a base of 
and a height of 
. Find the area of the parallelogram.
A parallelogram has a base of
By definition a parallelogram has two sets of opposite sides that are congruent/parallel. However, to find the area of a paralleogram, you need to know the base and height lengths. Since this problem provides both the base and height measurements, apply the formula:
The solution is:
By definition a parallelogram has two sets of opposite sides that are congruent/parallel. However, to find the area of a paralleogram, you need to know the base and height lengths. Since this problem provides both the base and height measurements, apply the formula:
The solution is:
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A parallelogram has a base of 
and a height measurement that is 
the base length. Find the area of the parallelogram.
A parallelogram has a base of
A parallelogram must have two sets of opposite sides that are congruent/parallel. However, to find the area of a paralleogram, you need to know the base and height lengths. Since this problem provides both the base and height measurements, apply the formula:
Before applying the formula you must find 
of 
.
The solution is:
A parallelogram must have two sets of opposite sides that are congruent/parallel. However, to find the area of a paralleogram, you need to know the base and height lengths. Since this problem provides both the base and height measurements, apply the formula:
Before applying the formula you must find
The solution is:
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Using the parallelogram shown above, find the area.
Using the parallelogram shown above, find the area.
By definition a parallelogram has two sets of opposite sides that are congruent/parallel. However to find the area of a paralleogram, you need to know the base and height lengths. Since this problem provides both the base and height measurements, apply the formula:
The solution is:
By definition a parallelogram has two sets of opposite sides that are congruent/parallel. However to find the area of a paralleogram, you need to know the base and height lengths. Since this problem provides both the base and height measurements, apply the formula:
The solution is:
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A parallelogram has a base of 
meters and a height measurement that is 
the base length. Find the area of the parallelogram.
A parallelogram has a base of
By definition a parallelogram has two sets of opposite sides that are congruent/parallel. However, to find the area of a paralleogram, you need to know the base and height lengths. Since this problem provides both the base and height measurements, apply the formula:
Before applying the formula you must find 
of 
.
The solution is:
By definition a parallelogram has two sets of opposite sides that are congruent/parallel. However, to find the area of a paralleogram, you need to know the base and height lengths. Since this problem provides both the base and height measurements, apply the formula:
Before applying the formula you must find
The solution is:
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A parallelogram has a base of 
and a height of 
. Find the area of the parallelogram.
A parallelogram has a base of
A parallelogram must have two sets of opposite sides that are congruent/parallel. However, to find the area of a paralleogram, you need to know the base and height lengths. Since this problem provides both the base and height measurements, apply the formula:
The solution is:
Note: prior to applying the formula, the answer choices require you to be able to convert 
to 
, as well as 
to 
. Or, you could have converted the mixed numbers to improper fractions and then multiplied the two terms:
A parallelogram must have two sets of opposite sides that are congruent/parallel. However, to find the area of a paralleogram, you need to know the base and height lengths. Since this problem provides both the base and height measurements, apply the formula:
The solution is:
Note: prior to applying the formula, the answer choices require you to be able to convert
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Find the area of the parallelogram shown above, excluding the interior space occupied by the blue rectangle.
Find the area of the parallelogram shown above, excluding the interior space occupied by the blue rectangle.
A parallelogram must have two sets of opposite sides that are congruent/parallel. However, to find the area of a paralleogram, you need to know the base and height lengths. Since this problem provides both the base and height measurements, apply the formula:
Additionally, this problem requires you to find the area of the interior rectangle. This can be simply found by applying the formula: 
Thus, the solution is:
A parallelogram must have two sets of opposite sides that are congruent/parallel. However, to find the area of a paralleogram, you need to know the base and height lengths. Since this problem provides both the base and height measurements, apply the formula:
Additionally, this problem requires you to find the area of the interior rectangle. This can be simply found by applying the formula:
Thus, the solution is:
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A parallelogram has a base of 
and a height measurement that is 
the base length. Find the area of the parallelogram.
A parallelogram has a base of
By definition a parallelogram has two sets of opposite sides that are congruent/parallel. However, to find the area of a paralleogram, you need to know the base and height lengths. Since this problem provides both the base and height measurements, apply the formula:
Before applying the formula you must find 
of 
.
The solution is:
Note: when working with multiples of ten remove zeros and then tack back onto the product.
There were two total zeros in the factors, so tack on two zeros to the product:
By definition a parallelogram has two sets of opposite sides that are congruent/parallel. However, to find the area of a paralleogram, you need to know the base and height lengths. Since this problem provides both the base and height measurements, apply the formula:
Before applying the formula you must find
The solution is:
Note: when working with multiples of ten remove zeros and then tack back onto the product.
There were two total zeros in the factors, so tack on two zeros to the product:
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Find the area for the parallelogram shown above.
Find the area for the parallelogram shown above.
A parallelogram must have two sets of opposite sides that are congruent/parallel. However, to find the area of a paralleogram, you need to know the base and height lengths. Since this problem provides both the base and height measurements, apply the formula:
The solution is:
A parallelogram must have two sets of opposite sides that are congruent/parallel. However, to find the area of a paralleogram, you need to know the base and height lengths. Since this problem provides both the base and height measurements, apply the formula:
The solution is:
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A parallelogram has a base of 
and a height measurement that is 
the base length. Find the area of the parallelogram.
A parallelogram has a base of
By definition a parallelogram has two sets of opposite sides that are congruent/parallel. However, to find the area of a paralleogram, you need to know the base and height lengths. Since this problem provides both the base and height measurements, apply the formula:
Before applying the formula you must find 
of 
.
The solution is:
Note: when working with multiples of ten remove zeros and then tack back onto the product.
There were two total zeros in the factors, so tack on two zeros to the product:
By definition a parallelogram has two sets of opposite sides that are congruent/parallel. However, to find the area of a paralleogram, you need to know the base and height lengths. Since this problem provides both the base and height measurements, apply the formula:
Before applying the formula you must find
The solution is:
Note: when working with multiples of ten remove zeros and then tack back onto the product.
There were two total zeros in the factors, so tack on two zeros to the product:
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ABCD is a rectangle.
Quantity A: The area of AEB
Quantity B: The area BEC
ABCD is a rectangle.
Quantity A: The area of AEB
Quantity B: The area BEC
The area of a triangle is 
Consider the rectangle ABCD
As a rectangle:
With 
appearing directly in the center of the rectangle.
The area of 
Notice that the 
term corresponds to the triangle's height.
The area of 
The two quantities are equal.
The area of a triangle is
Consider the rectangle ABCD
As a rectangle:
With
The area of
Notice that the
The area of
The two quantities are equal.
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If a rectangle's width is increased by 20%, and its length is decreased by 20%, which statement most accurately reflects the rectangle's change in area?
If a rectangle's width is increased by 20%, and its length is decreased by 20%, which statement most accurately reflects the rectangle's change in area?
If l x w = A, then we will call the new area B. B results when l has been decreased by 20% and w has been increased by 20%: in other words, 80% of l and 120% of w.
This is expressed as (0.8) l * (1.2) w = B
The simplest way to determine the result is to create rectangles with actual integers and see what happens. Let's say our first rectangle is a square 10 units by 10 units.
10 * 10 = 100 for our A.
Plugging these numbers into our new equation gives us:
(0.8)10 * (1.2)10 = 96.
So the final step is to compare the areas. Because we used a 10 x 10 rectangle as our example, this step will be easy, since 100% = 100.
96/100 = 0.96
0.96 * 100 = 96%
Which reflects a 4% reduction in the size of the rectangle.
Any other length and width used for this rectangle will result in a 4% reduction in size when the above parameters are applied.
If l x w = A, then we will call the new area B. B results when l has been decreased by 20% and w has been increased by 20%: in other words, 80% of l and 120% of w.
This is expressed as (0.8) l * (1.2) w = B
The simplest way to determine the result is to create rectangles with actual integers and see what happens. Let's say our first rectangle is a square 10 units by 10 units.
10 * 10 = 100 for our A.
Plugging these numbers into our new equation gives us:
(0.8)10 * (1.2)10 = 96.
So the final step is to compare the areas. Because we used a 10 x 10 rectangle as our example, this step will be easy, since 100% = 100.
96/100 = 0.96
0.96 * 100 = 96%
Which reflects a 4% reduction in the size of the rectangle.
Any other length and width used for this rectangle will result in a 4% reduction in size when the above parameters are applied.
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