Geometry - GRE Quantitative Reasoning
Card 1 of 2800
Which line is perpendicular to the line between the points (22,24) and (31,4)?
Which line is perpendicular to the line between the points (22,24) and (31,4)?
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The line will be perpendicular if the slope is the negative reciprocal.
First we need to find the slope of our line between points (22,24) and (31,4). Slope = rise/run = (24 – 4)/(22 – 31) = 20/–9 = –2.22.
The negative reciprocal of this must be a positive fraction, so we can eliminate y = –3_x_ + 5 (because the slope is negative).
The negative reciprocal of –2.22, and therefore the slope of the perpendicular line, will be –1/–2.22 = .45, so we can also eliminate y = x (slope of 1).
Now let's look at the line between points (9, 5) and (48, 19). This slope = (5 – 19)/(9 – 48) = .358, which is incorrect.
The next answer choice is y = .45_x_ + 10. The slope is .45, which is what we're looking for so this is the correct answer.
To double check, the last answer choice is the line between (4, 7) and (7, 4). This slope = (7 – 4) / (4 – 7) = –1, which is also incorrect.
The line will be perpendicular if the slope is the negative reciprocal.
First we need to find the slope of our line between points (22,24) and (31,4). Slope = rise/run = (24 – 4)/(22 – 31) = 20/–9 = –2.22.
The negative reciprocal of this must be a positive fraction, so we can eliminate y = –3_x_ + 5 (because the slope is negative).
The negative reciprocal of –2.22, and therefore the slope of the perpendicular line, will be –1/–2.22 = .45, so we can also eliminate y = x (slope of 1).
Now let's look at the line between points (9, 5) and (48, 19). This slope = (5 – 19)/(9 – 48) = .358, which is incorrect.
The next answer choice is y = .45_x_ + 10. The slope is .45, which is what we're looking for so this is the correct answer.
To double check, the last answer choice is the line between (4, 7) and (7, 4). This slope = (7 – 4) / (4 – 7) = –1, which is also incorrect.
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Which best describes the relationship between the lines 
and 
?
Which best describes the relationship between the lines
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We first need to recall the following relationships:
Lines with the same slope and same 
-intercept are really the same line.
Lines with the same slope and different 
-intercepts are parallel.
Lines with slopes that are negative reciprocals are perpendicular.
Then we identify the slopes of the two lines by comparing the equations to the slope-intercept form 
, where 
is the slope and 
is the 
-intercept. By inspection we see the lines have slopes of 
and 
. Since these are different, the "parallel" and "same line" choices are eliminated. To test if the slopes are negative reciprocals, we take one of the slopes, change its sign, and flip it upside-down. Starting with 
and changing the sign gives 
, then flipping gives 
. This is the same as the slope of the second line, so the two slopes are negative reciprocals and the lines are perpendicular.
We first need to recall the following relationships:
Lines with the same slope and same
Lines with the same slope and different
Lines with slopes that are negative reciprocals are perpendicular.
Then we identify the slopes of the two lines by comparing the equations to the slope-intercept form
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There is a line defined by two end-points, 
and 
. The midpoint between these two points is 
. What is the value of the point 
?
There is a line defined by two end-points,
Tap to reveal answer
Recall that to find the midpoint of two points 
and 
, you use the equation:
.
(It is just like finding the average of the two points, really.)
So, for our equation, we know the following:
You merely need to solve each coordinate for its respective value.
Then, for the y-coordinate:
Therefore, our other point is: 
Recall that to find the midpoint of two points
(It is just like finding the average of the two points, really.)
So, for our equation, we know the following:
You merely need to solve each coordinate for its respective value.
Then, for the y-coordinate:
Therefore, our other point is:
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There is a line defined by two end-points, 
and 
. The midpoint between these two points is 
. What is the value of the point 
?
There is a line defined by two end-points,
Tap to reveal answer
Recall that to find the midpoint of two points 
and 
, you use the equation:
.
(It is just like finding the average of the two points, really.)
So, for our equation, we know the following:
You merely need to solve each coordinate for its respective value.
Then, for the y-coordinate:
Therefore, our other point is: 
Recall that to find the midpoint of two points
(It is just like finding the average of the two points, really.)
So, for our equation, we know the following:
You merely need to solve each coordinate for its respective value.
Then, for the y-coordinate:
Therefore, our other point is:
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Which of the following equations represents a line that is perpendicular to the line with points 
and 
?
Which of the following equations represents a line that is perpendicular to the line with points
Tap to reveal answer
If lines are perpendicular, then their slopes will be negative reciprocals.
First, we need to find the slope of the given line.
Because we know that our given line's slope is 
, the slope of the line perpendicular to it must be 
.
If lines are perpendicular, then their slopes will be negative reciprocals.
First, we need to find the slope of the given line.
Because we know that our given line's slope is
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Which of the following lines is perpindicular to 
Which of the following lines is perpindicular to
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When determining if a two lines are perpindicular, we are only concerned about their slopes. Consider the basic equation of a line, 
, where m is the slope of the line. Two lines are perpindicular to each other if one slope is the negative and reciprocal of the other.
The first step of this problem is to get it into the form, 
, which is 
. Now we know that the slope, m, is 
. The reciprocal of that is 
, and the negative of that is 
. Therefore, any line that has a slope of 
will be perpindicular to the original line.
When determining if a two lines are perpindicular, we are only concerned about their slopes. Consider the basic equation of a line,
The first step of this problem is to get it into the form,
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Quantitative Comparison
Quantity A: The degree measure of any angle in an equilateral triangle
Quantity B: The degree measure of any angle in a regular hexagon
Quantitative Comparison
Quantity A: The degree measure of any angle in an equilateral triangle
Quantity B: The degree measure of any angle in a regular hexagon
Tap to reveal answer
We know the three angles in a triangle add up to 180 degrees, and all three angles are 60 degrees in an equilateral triangle.
A hexagon has six sides, and we can use the formula degrees = (# of sides – 2) * 180. Then degrees = (6 – 2) * 180 = 720 degrees. Each angle is 720/6 = 120 degrees.
Quantity B is greater.
We know the three angles in a triangle add up to 180 degrees, and all three angles are 60 degrees in an equilateral triangle.
A hexagon has six sides, and we can use the formula degrees = (# of sides – 2) * 180. Then degrees = (6 – 2) * 180 = 720 degrees. Each angle is 720/6 = 120 degrees.
Quantity B is greater.
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Quantity A: Double the measure of a single interior angle of an equilateral triangle.
Quantity B: The measure of a single interior angle of a hexagon.
Quantity A: Double the measure of a single interior angle of an equilateral triangle.
Quantity B: The measure of a single interior angle of a hexagon.
Tap to reveal answer
Begin with Quantity A. We know the measure of one angle in an equilateral triangle is 60. Therefore, double the angle is 120 degrees.
For the hexagon, use the formula for the sum of the interior angles:
where n= number of sides in a regular polygon
If the sum of the interior angles of a regular hexagon is 
degrees, then one angle is 
degrees.
The two quantities are equal.
Begin with Quantity A. We know the measure of one angle in an equilateral triangle is 60. Therefore, double the angle is 120 degrees.
For the hexagon, use the formula for the sum of the interior angles:
where n= number of sides in a regular polygon
If the sum of the interior angles of a regular hexagon is
The two quantities are equal.
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Quantitative Comparison
Quantity A: The degree measure of any angle in an equilateral triangle
Quantity B: The degree measure of any angle in a regular hexagon
Quantitative Comparison
Quantity A: The degree measure of any angle in an equilateral triangle
Quantity B: The degree measure of any angle in a regular hexagon
Tap to reveal answer
We know the three angles in a triangle add up to 180 degrees, and all three angles are 60 degrees in an equilateral triangle.
A hexagon has six sides, and we can use the formula degrees = (# of sides – 2) * 180. Then degrees = (6 – 2) * 180 = 720 degrees. Each angle is 720/6 = 120 degrees.
Quantity B is greater.
We know the three angles in a triangle add up to 180 degrees, and all three angles are 60 degrees in an equilateral triangle.
A hexagon has six sides, and we can use the formula degrees = (# of sides – 2) * 180. Then degrees = (6 – 2) * 180 = 720 degrees. Each angle is 720/6 = 120 degrees.
Quantity B is greater.
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Quantity A: Double the measure of a single interior angle of an equilateral triangle.
Quantity B: The measure of a single interior angle of a hexagon.
Quantity A: Double the measure of a single interior angle of an equilateral triangle.
Quantity B: The measure of a single interior angle of a hexagon.
Tap to reveal answer
Begin with Quantity A. We know the measure of one angle in an equilateral triangle is 60. Therefore, double the angle is 120 degrees.
For the hexagon, use the formula for the sum of the interior angles:
where n= number of sides in a regular polygon
If the sum of the interior angles of a regular hexagon is degrees, then one angle is degrees.
The two quantities are equal.
Begin with Quantity A. We know the measure of one angle in an equilateral triangle is 60. Therefore, double the angle is 120 degrees.
For the hexagon, use the formula for the sum of the interior angles:
where n= number of sides in a regular polygon
If the sum of the interior angles of a regular hexagon is
The two quantities are equal.
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Quantitative Comparison
Quantity A: The degree measure of any angle in an equilateral triangle
Quantity B: The degree measure of any angle in a regular hexagon
Quantitative Comparison
Quantity A: The degree measure of any angle in an equilateral triangle
Quantity B: The degree measure of any angle in a regular hexagon
Tap to reveal answer
We know the three angles in a triangle add up to 180 degrees, and all three angles are 60 degrees in an equilateral triangle.
A hexagon has six sides, and we can use the formula degrees = (# of sides – 2) * 180. Then degrees = (6 – 2) * 180 = 720 degrees. Each angle is 720/6 = 120 degrees.
Quantity B is greater.
We know the three angles in a triangle add up to 180 degrees, and all three angles are 60 degrees in an equilateral triangle.
A hexagon has six sides, and we can use the formula degrees = (# of sides – 2) * 180. Then degrees = (6 – 2) * 180 = 720 degrees. Each angle is 720/6 = 120 degrees.
Quantity B is greater.
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Quantity A: Double the measure of a single interior angle of an equilateral triangle.
Quantity B: The measure of a single interior angle of a hexagon.
Quantity A: Double the measure of a single interior angle of an equilateral triangle.
Quantity B: The measure of a single interior angle of a hexagon.
Tap to reveal answer
Begin with Quantity A. We know the measure of one angle in an equilateral triangle is 60. Therefore, double the angle is 120 degrees.
For the hexagon, use the formula for the sum of the interior angles:
where n= number of sides in a regular polygon
If the sum of the interior angles of a regular hexagon is degrees, then one angle is degrees.
The two quantities are equal.
Begin with Quantity A. We know the measure of one angle in an equilateral triangle is 60. Therefore, double the angle is 120 degrees.
For the hexagon, use the formula for the sum of the interior angles:
where n= number of sides in a regular polygon
If the sum of the interior angles of a regular hexagon is
The two quantities are equal.
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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?
Tap to reveal 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.
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
Tap to reveal answer
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?
Tap to reveal answer
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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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?
Tap to reveal 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.
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
Tap to reveal answer
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?
Tap to reveal answer
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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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?
Tap to reveal 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.
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
Tap to reveal answer
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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