Geometry - GMAT Quantitative
Card 0 of 5403
Define a function 
as follows:
for nonzero real numbers 
.
Where is the vertical asymptote of the graph of 
in relation to the 
-axis - is it to the left of it, to the right of it, or on it?
Statement 1: 
and 
are both positive.
Statement 2: 
and 
are of opposite sign.
Define a function
for nonzero real numbers
Where is the vertical asymptote of the graph of
Statement 1:
Statement 2:
Since only positive numbers have logarithms, the expression 
must be positive, so
Therefore, the vertical asymptote must be the vertical line of the equation
.
In order to determine which side of the 
-axis the vertical asymptote falls, it is necessary to find the sign of 
; if it is negative, it is on the left side, if it is positive, it is on the right side.
Assume both statements are true. By Statement 1, 
is positive. If 
is positive, then 
is negative, and vice versa. However, Statement 2, which mentions 
, does not give its actual sign - just the fact that its sign is the opposite of that of 
, which we are not given either. The two statements therefore give insufficient information.
Since only positive numbers have logarithms, the expression
Therefore, the vertical asymptote must be the vertical line of the equation
In order to determine which side of the
Assume both statements are true. By Statement 1,
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Triangle 
has height 
. What is the length of 
?
(1) 
.
(2) 
.
Triangle
(1)
(2)
Since we don't know what type the triangle is, we would not only need information about the lengths of the side but also about the characteristics of the triangle.
Statement 1 gives us the length of a side. However, we can't do anything, since we don't know the length of DC, which would allow us the know BD with the Pythagorean Theorem.
Statement 2 also only gives us information about one side of the triangle. Alone it doesn't allow us to calculate any other length.
Even taken together these statements are insufficient since, we don't know any pair of lengths to use in the Pythagorean Theorem. Even though the triangle looks like a isosceles triangle, it doesn't mean that it is.
Since we don't know what type the triangle is, we would not only need information about the lengths of the side but also about the characteristics of the triangle.
Statement 1 gives us the length of a side. However, we can't do anything, since we don't know the length of DC, which would allow us the know BD with the Pythagorean Theorem.
Statement 2 also only gives us information about one side of the triangle. Alone it doesn't allow us to calculate any other length.
Even taken together these statements are insufficient since, we don't know any pair of lengths to use in the Pythagorean Theorem. Even though the triangle looks like a isosceles triangle, it doesn't mean that it is.
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is a triangle with height 
. What is the length 
?
(1) The triangle has an area of 
and 
.
(2) 
and 
.
(1) The triangle has an area of
(2)
To find the length DC, we need to know AD and AC or any of those two provided that ABC is isosceles or equilateral.
Statement 1 tells us the area of the triangle with information about a part of side AC. Since we don't know properties of the triangle, these other lengths can vest many values, just AC can be 12, 24 or 48. Therefore we don't have enough information.
Statement 2 gives us information about angles of the triangle. From what we are told we can see that the triangle is isosceles. Indeed, we know that 
since BD is the height. Therefore 
. Now, that we know that the triangle is isosceles, we know that AC must be 12, since D is the midpoint of AC. Therefore DC must be 6.
Hence, both statements taken together are sufficient.
To find the length DC, we need to know AD and AC or any of those two provided that ABC is isosceles or equilateral.
Statement 1 tells us the area of the triangle with information about a part of side AC. Since we don't know properties of the triangle, these other lengths can vest many values, just AC can be 12, 24 or 48. Therefore we don't have enough information.
Statement 2 gives us information about angles of the triangle. From what we are told we can see that the triangle is isosceles. Indeed, we know that
Hence, both statements taken together are sufficient.
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Find the hypotenuse of an obtuse triangle.
Statement 1: Two given lengths with an inscribed angle.
Statement 2: Two known angles.
Find the hypotenuse of an obtuse triangle.
Statement 1: Two given lengths with an inscribed angle.
Statement 2: Two known angles.
Statement 1: Two given lengths with an inscribed angle.
Draw a picture of the scenario. The values of 
, 
, and angle 
are known values.
Use the Law of Cosines to determine side length 
.
Statement 2: Two known angles.
There is insufficient information to solve for the length of the hypotenuse with only two interior angles. The third angle can be determined by subtracting the 2 angles from 180 degrees.
The triangle can be enlarged or shrunk to any degree with any scale factor and still yield the same interior angles. There must also be at least 1 side length in order to calculate the hypotenuse of the triangle by the Law of Cosines.
Therefore:
Statement 1: Two given lengths with an inscribed angle.
Draw a picture of the scenario. The values of
Use the Law of Cosines to determine side length
Statement 2: Two known angles.
There is insufficient information to solve for the length of the hypotenuse with only two interior angles. The third angle can be determined by subtracting the 2 angles from 180 degrees.
The triangle can be enlarged or shrunk to any degree with any scale factor and still yield the same interior angles. There must also be at least 1 side length in order to calculate the hypotenuse of the triangle by the Law of Cosines.
Therefore:
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Define a function 
as follows:
for nonzero real numbers 
.
Give the equation of the vertical asymptote of the graph of 
.
Statement 1: 
Statement 2: 
Define a function
for nonzero real numbers
Give the equation of the vertical asymptote of the graph of
Statement 1:
Statement 2:
Since only positive numbers have logarithms,
Therefore, the vertical asymptote must be the vertical line of the equation
.
Assume both statements to be true. We need two numbers 
and 
whose sum is 7 and whose product is 12; by trial and error, we can find these numbers to be 3 and 4. However, without further information, we have no way of determining which of 
and 
is 3 and which is 4, so the asymptote can be either 
or 
.
Since only positive numbers have logarithms,
Therefore, the vertical asymptote must be the vertical line of the equation
Assume both statements to be true. We need two numbers
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Define a function 
as follows:
for nonzero real numbers 
.
What is the equation of the vertical asymptote of the graph of 
?
Statement 1: 
and 
are of opposite sign.
Statement 2: 
Define a function
for nonzero real numbers
What is the equation of the vertical asymptote of the graph of
Statement 1:
Statement 2:
Since only positive numbers have logarithms,
Therefore, the vertical asymptote must be the vertical line of the equation
.
In order to determine which side of the 
-axis the vertical asymptote falls, it is necessary to find the sign of 
; if it is negative, it is on the left side, and if it is positive, it is on the right side.
Statement 1 alone only gives us that 
is a different sign from 
; without any information about the sign of 
, we cannot answer the question.
Statement 2 alone gives us that 
, and, consequently, 
. This means that 
and 
are of opposite sign. But again, with no information about the sign of 
, we cannot answer the question.
Assume both statements to be true. Since, from the two statements, both 
and 
are of the opposite sign from 
, 
and 
are of the same sign. Their quotient 
is positive, and 
is negative, so the vertical asymptote 
is to the left of the 
-axis.
Since only positive numbers have logarithms,
Therefore, the vertical asymptote must be the vertical line of the equation
In order to determine which side of the
Statement 1 alone only gives us that
Statement 2 alone gives us that
Assume both statements to be true. Since, from the two statements, both
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Define a function 
as follows:
for nonzero real numbers 
.
Where is the vertical asymptote of the graph of 
in relation to the 
-axis - is it to the left of it, to the right of it, or on it?
Statement 1: 
Statement 2: 
Define a function
for nonzero real numbers
Where is the vertical asymptote of the graph of
Statement 1:
Statement 2:
Only positive numbers have logarithms, so:
Therefore, the vertical asymptote must be the vertical line of the equation
.
In order to determine which side of the 
-axis the vertical asymptote falls, it is necessary to find out whether the signs of 
and 
are the same or different. If 
and 
are of the same sign, then their quotient 
is positive, and 
is negative, putting 
on the left side of the 
-axis. If 
and 
are of different sign, then their quotient 
is negative, and 
is positive, putting 
on the right side of the 
-axis.
Statement 1 alone does not give us enough information to determine whether 
and 
have different signs. 
, for example, but 
, also.
From Statement 2, since the product of 
and 
is negative, they must be of different sign. Therefore, 
is positive, and 
falls to the right of the 
-axis.
Only positive numbers have logarithms, so:
Therefore, the vertical asymptote must be the vertical line of the equation
In order to determine which side of the
Statement 1 alone does not give us enough information to determine whether
From Statement 2, since the product of
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In which quadrant is the point 
located: I, II, III, or IV?
Statement 1: 
Statement 2: 
In which quadrant is the point
Statement 1:
Statement 2:
Assume Statement 1 alone. The set of points that satisfy the equation is the set of all points on the line of the equation
which will pass through at least two quadrants on the coordinate plane. Therefore, Statement 1 provides insufficient information.
Now assume Statement 2 alone. The set of points that satisfy the equation is the set of all points of the circle of the equation
This circle has 
as its center and 
as its radius. Since its center is 
, which is 5 units away from its closest axis, and the radius is less than 5 units, the circle never intersects an axis, so it is contained entirely within the same quadrant as its center. The center has negative 
- and 
-coordinates, placing it, and the entire circle, in Quadrant III.
Assume Statement 1 alone. The set of points that satisfy the equation is the set of all points on the line of the equation
which will pass through at least two quadrants on the coordinate plane. Therefore, Statement 1 provides insufficient information.
Now assume Statement 2 alone. The set of points that satisfy the equation is the set of all points of the circle of the equation
This circle has
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In which quadrant is the point 
located: I, II, III, or IV?
Statement 1: 
Statement 2: 
In which quadrant is the point
Statement 1:
Statement 2:
Assume Statement 1 alone. The set of points that satisfy the equation is the set of all points on the line of the equation
,
which will pass through at least two quadrants on the coordinate plane. Therefore, Statement 1 provides insufficient information. By the same argument, Statement 2 is also insuffcient.
Now assume both statements to be true. The two statements together form a system of linear equations which can be solved using the elimination method:
Now, substitute back:
The point is 
, which has a positive 
-coordinate and a negative 
-coordinate and is consequently in Quadrant IV.
Assume Statement 1 alone. The set of points that satisfy the equation is the set of all points on the line of the equation
which will pass through at least two quadrants on the coordinate plane. Therefore, Statement 1 provides insufficient information. By the same argument, Statement 2 is also insuffcient.
Now assume both statements to be true. The two statements together form a system of linear equations which can be solved using the elimination method:
Now, substitute back:
The point is
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How many sides does a regular polygon have?
-
Each of the angles measures 140 degrees.
-
Each of the sides has measure 8.
How many sides does a regular polygon have?
-
Each of the angles measures 140 degrees.
-
Each of the sides has measure 8.
The relationship between the number of sides of a regular polygon 
and the measure of a single angle 
is
If we are given that 
, then we can substitute and solve for 
:
making the figure a nine-sided polygon.
Knowing only the measure of each side is neither necessary nor helpful; for example, it is possible to construct an equilateral triangle with sidelength 8 or a square with sidelength 8.
The correct answer is that Statement 1 alone is sufficient to answer the question, but not Statement 2 alone.
The relationship between the number of sides of a regular polygon
If we are given that
making the figure a nine-sided polygon.
Knowing only the measure of each side is neither necessary nor helpful; for example, it is possible to construct an equilateral triangle with sidelength 8 or a square with sidelength 8.
The correct answer is that Statement 1 alone is sufficient to answer the question, but not Statement 2 alone.
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Note: Figure NOT drawn to scale.
Is the pentagon in the diagram above a regular pentagon?
Statement 1: 
Statement 2: The hexagon in the diagram is not a regular hexagon.
Note: Figure NOT drawn to scale.
Is the pentagon in the diagram above a regular pentagon?
Statement 1:
Statement 2: The hexagon in the diagram is not a regular hexagon.
Information about the hexagon is irrelevant, so Statement 2 has no bearing on the answer to the question.
The measures of the exterior angles of any pentagon, one per vertex, total 
, and they are congruent if the pentagon is regular, so if this is the case, each would measure 
. But if Statement 1 is true, then an exterior angle of the pentagon measures 
. Therefore, Statement 1 is enough to answer the question in the negative.
Information about the hexagon is irrelevant, so Statement 2 has no bearing on the answer to the question.
The measures of the exterior angles of any pentagon, one per vertex, total
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Note: Figure NOT drawn to scale.
What is 
?
Statement 1: 
, 
, and 
are the first three terms, in order, of an arithmetic sequence.
Statement 2: 
Note: Figure NOT drawn to scale.
What is
Statement 1:
Statement 2:
The sum of the measures of the angles of a hexagon is 
Therefore,
Suppose we only know that 
, 
, and 
are the first three terms of an arithmetic sequence, in order. Then for some common difference 
,
Suppose we only know that 
. Then
With no further information, we cannot determine 
.
The sum of the measures of the angles of a hexagon is
Therefore,
Suppose we only know that
Suppose we only know that
With no further information, we cannot determine
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Note: Figure NOT drawn to scale.
The diagram above shows a triangle and a rhombus sharing a side. Is that rhombus a square?
Statement 1: The triangle is not equilateral.
Statement 2: 
Note: Figure NOT drawn to scale.
The diagram above shows a triangle and a rhombus sharing a side. Is that rhombus a square?
Statement 1: The triangle is not equilateral.
Statement 2:
To show that the rhombus is a square, you need to demonstrate that one of its angles is a right angle - that is, 
. Both statements together are insufficent - if 
, you would need to demonstrate that 
is true or false, and the fact that the triangle is not equilateral is not enough to prove or to disprove this.
To show that the rhombus is a square, you need to demonstrate that one of its angles is a right angle - that is,
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What is the measure of an interior angle of a regular polygon?
Statement 1: The polygon has 20 sides.
Statement 2: An exterior angle of the polygon measures 
.
What is the measure of an interior angle of a regular polygon?
Statement 1: The polygon has 20 sides.
Statement 2: An exterior angle of the polygon measures
From Statement 1, you can calculate the measure of an interior angle as follows:
From Statement 2, since an interior angle and an exterior angle at the same vertex form a linear pair, they are supplementary, so you can subtract 18 from 180:
From Statement 1, you can calculate the measure of an interior angle as follows:
From Statement 2, since an interior angle and an exterior angle at the same vertex form a linear pair, they are supplementary, so you can subtract 18 from 180:
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Is 
a regular pentagon?
Statement 1: 
Statement 2: 
Is
Statement 1:
Statement 2:
All of the interior angles of a regular polygon are congruent, as are all of its sides. Statement 1 violates the former condition; statement 2 violates the latter.
All of the interior angles of a regular polygon are congruent, as are all of its sides. Statement 1 violates the former condition; statement 2 violates the latter.
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Is the degree measure of an exterior angle of a regular polygon an integer?
Statement 1: The number of sides of the polygon is divisible by 7.
Statement 2: The number of sides of the polygon is divisible by 10.
Is the degree measure of an exterior angle of a regular polygon an integer?
Statement 1: The number of sides of the polygon is divisible by 7.
Statement 2: The number of sides of the polygon is divisible by 10.
The sum of the degree measures of the exterior angles, one per vertex, of any polygon is 360, so each exterior angle of a regular polygon with 
sides measures 
.
For 
to be an integer, every factor of 
must be a factor of 360. This does not happen if 7 is a factor of 
, so Statement 1 disproves this. This may or may not happen if 10 is a factor of 
- 
, but 
, so Statement 2 does not provide an answer.
The sum of the degree measures of the exterior angles, one per vertex, of any polygon is 360, so each exterior angle of a regular polygon with
For
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Is the degree measure of an exterior angle of a regular polygon an integer?
Statement 1: The number of sides of the polygon is a factor of 30.
Statement 2: The number of sides of the polygon is a factor of 40.
Is the degree measure of an exterior angle of a regular polygon an integer?
Statement 1: The number of sides of the polygon is a factor of 30.
Statement 2: The number of sides of the polygon is a factor of 40.
The sum of the degree measures of the exterior angles, one per vertex, of any polygon is 360, so each exterior angle of a regular polygon with 
sides measures 
. Therefore, the measure of one exterior angle of a regular polygon is an integer if and only if 
is a factor of 360.
If 
is a factor of a factor of 360, however, then 
is a factor of 360. 30 and 40 are both factors of 360: 
and 
. Therefore, it follows from either statement that the number of sides is a factor of 360, and each exterior angle has a degree measure that is an integer.
The sum of the degree measures of the exterior angles, one per vertex, of any polygon is 360, so each exterior angle of a regular polygon with
If
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What is the measure of 
?
Statement 1: is an exterior angle of an equilateral triangle.
Statement 2: is an interior angle of a regular hexagon.
What is the measure of
Statement 1:
Statement 2:
An exterior angle of an equilateral triangle measures 
. An interior angle of a regular hexagon measures 
. Either statement is sufficient.
An exterior angle of an equilateral triangle measures
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A nonagon is a nine-sided polygon.
Is Nonagon 
regular?
Statement 1: 
Statement 2: 
A nonagon is a nine-sided polygon.
Is Nonagon
Statement 1:
Statement 2:
Each angle of a regular nonagon measures 
Therefore, each of the two statements proves that the nonagon is not regular.
Each angle of a regular nonagon measures
Therefore, each of the two statements proves that the nonagon is not regular.
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Note: Figure NOT drawn to scale.
Refer to the above figure. Is Pentagon 
regular?
Statement 1: 
Statement 2: 
Note: Figure NOT drawn to scale.
Refer to the above figure. Is Pentagon
Statement 1:
Statement 2:
If a shape is regular, that means that all of its sides are equal. It also means that all of its interior angles are equal. Finally, if all of the interior angles are equal, then the exterior angles will all be equal to each other as well.
In a regular polygon, we can find the measure of ANY exterior angle by using the formula
where 
is equal to the number of sides.
Each exterior angle of a regular (five-sided) pentagon measures
Statement 1 alone neither proves nor disproves that the pentagon is regular. We now know that one exterior angle is 
, but we do not know if any of the other exterior angles are also 
.
Statement 2, however, proves that the pentagon is not regular, as it has at least one exterior angle that does not have measure 
.
If a shape is regular, that means that all of its sides are equal. It also means that all of its interior angles are equal. Finally, if all of the interior angles are equal, then the exterior angles will all be equal to each other as well.
In a regular polygon, we can find the measure of ANY exterior angle by using the formula
where
Each exterior angle of a regular (five-sided) pentagon measures
Statement 1 alone neither proves nor disproves that the pentagon is regular. We now know that one exterior angle is
Statement 2, however, proves that the pentagon is not regular, as it has at least one exterior angle that does not have measure
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