Geometry - GRE Quantitative Reasoning
Card 1 of 2800
What is the slope of the line represented by the equation 
?
What is the slope of the line represented by the equation
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To rearrange the equation into a 
format, you want to isolate the 
so that it is the sole variable, without a coefficient, on one side of the equation.
First, add 
to both sides to get 
.
Then, divide both sides by 6 to get 
.
If you divide each part of the numerator by 6, you get 
. This is in a 
form, and the 
is equal to 
, which is reduced down to 
for the correct answer.
To rearrange the equation into a
First, add
Then, divide both sides by 6 to get
If you divide each part of the numerator by 6, you get
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What is the slope of the line 
?
What is the slope of the line
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To find the slope, put the equation in slope-intercept form 
. In this case we have 
, which indicates that the slope is 
.
To find the slope, put the equation in slope-intercept form
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What is the distance between 
and 
?
What is the distance between
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distance2 = (_x_2 – _x_1)2 + (_y_2 – _y_1)2 + (_z_2 – _z_1)2
= (4 – 2)2 + (6 – 3)2 + (5 – 4)2
= 22 + 32 + 12
= 14
distance = √14
distance2 = (_x_2 – _x_1)2 + (_y_2 – _y_1)2 + (_z_2 – _z_1)2
= (4 – 2)2 + (6 – 3)2 + (5 – 4)2
= 22 + 32 + 12
= 14
distance = √14
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What is the slope of a line running through points 
and 
?
What is the slope of a line running through points
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The slope is equal to the difference between the y-coordinates divided by the difference between the x-coordinates.
Use the give points in this formula to calculate the slope.
The slope is equal to the difference between the y-coordinates divided by the difference between the x-coordinates.
Use the give points in this formula to calculate the slope.
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What is the slope of the equation 
?
What is the slope of the equation
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To find the slope of a line, you should convert an equation to the slope-intercept form. In this case, the equation would be 
, which means the slope is 
.
To find the slope of a line, you should convert an equation to the slope-intercept form. In this case, the equation would be
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What is the slope of a line defined by the equation:
What is the slope of a line defined by the equation:
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A question like this is actually rather easy. All you need to do is rewrite the equation in slope intercept form, that is:
Therefore, begin to simplify:
Becomes...
Then...
Finally, divide both sides by 
:
The coefficient for the 
term is your slope: 
A question like this is actually rather easy. All you need to do is rewrite the equation in slope intercept form, that is:
Therefore, begin to simplify:
Becomes...
Then...
Finally, divide both sides by
The coefficient for the
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Which of the following could be an equation for the red line pictured above?
Which of the following could be an equation for the red line pictured above?
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There are two key facts to register about this drawing. First, the line clearly has a negative slope, given that it runs "downhill" when you look at it from left to right. Secondly, it has a positive y-intercept. Therefore, you know that the coefficient for the 
term must be negative, and the numerical coefficient for the y-intercept must be positive. This only occurs in the equation 
. Therefore, this is the only viable option.
There are two key facts to register about this drawing. First, the line clearly has a negative slope, given that it runs "downhill" when you look at it from left to right. Secondly, it has a positive y-intercept. Therefore, you know that the coefficient for the
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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,
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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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Quantitative Comparison
Column A
Area
Column B
Perimeter
Quantitative Comparison
Column A
Area
Column B
Perimeter
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To find the perimeter, add up the sides, here 5 + 12 + 13 = 30. To find the area, multiply the two legs together and divide by 2, here (5 * 12)/2 = 30.
To find the perimeter, add up the sides, here 5 + 12 + 13 = 30. To find the area, multiply the two legs together and divide by 2, here (5 * 12)/2 = 30.
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Given triangle ACE where B is the midpoint of AC, what is the area of triangle ABD?
Given triangle ACE where B is the midpoint of AC, what is the area of triangle ABD?
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If B is a midpoint of AC, then we know AB is 12. Moreover, triangles ACE and ABD share angle DAB and have right angles which makes them similar triangles. Thus, their sides will all be proportional, and BD is 4. 1/2bh gives us 1/2 * 12 * 4, or 24.
If B is a midpoint of AC, then we know AB is 12. Moreover, triangles ACE and ABD share angle DAB and have right angles which makes them similar triangles. Thus, their sides will all be proportional, and BD is 4. 1/2bh gives us 1/2 * 12 * 4, or 24.
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What is the area of a right triangle with hypotenuse of 13 and base of 12?
What is the area of a right triangle with hypotenuse of 13 and base of 12?
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Area = 1/2(base)(height). You could use Pythagorean theorem to find the height or, if you know the special right triangles, recognize the 5-12-13. The area = 1/2(12)(5) = 30.
Area = 1/2(base)(height). You could use Pythagorean theorem to find the height or, if you know the special right triangles, recognize the 5-12-13. The area = 1/2(12)(5) = 30.
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Quantitative Comparison
Quantity A: the area of a right triangle with sides 10, 24, 26
Quantity B: twice the area of a right triangle with sides 5, 12, 13
Quantitative Comparison
Quantity A: the area of a right triangle with sides 10, 24, 26
Quantity B: twice the area of a right triangle with sides 5, 12, 13
Tap to reveal answer
Quantity A: area = base * height / 2 = 10 * 24 / 2 = 120
Quantity B: 2 * area = 2 * base * height / 2 = base * height = 5 * 12 = 60
Therefore Quantity A is greater.
Quantity A: area = base * height / 2 = 10 * 24 / 2 = 120
Quantity B: 2 * area = 2 * base * height / 2 = base * height = 5 * 12 = 60
Therefore Quantity A is greater.
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Quantitative Comparison
Quantity A: The area of a triangle with a height of 6 and a base of 7
Quantity B: Half the area of a trapezoid with a height of 6, a base of 6, and another base of 10
Quantitative Comparison
Quantity A: The area of a triangle with a height of 6 and a base of 7
Quantity B: Half the area of a trapezoid with a height of 6, a base of 6, and another base of 10
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Quantity A: Area = 1/2 * b * h = 1/2 * 6 * 7 = 42/2 = 21
Quantity B: Area = 1/2 * (_b_1 + _b_2) * h = 1/2 * (6 + 10) * 6 = 48
Half of the area = 48/2 = 24
Quantity B is greater.
Quantity A: Area = 1/2 * b * h = 1/2 * 6 * 7 = 42/2 = 21
Quantity B: Area = 1/2 * (_b_1 + _b_2) * h = 1/2 * (6 + 10) * 6 = 48
Half of the area = 48/2 = 24
Quantity B is greater.
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Consider the lines described by the following two equations:
4y = 3x2
3y = 4x2
Find the vertical distance between the two lines at the points where x = 6.
Consider the lines described by the following two equations:
4y = 3x2
3y = 4x2
Find the vertical distance between the two lines at the points where x = 6.
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Since the vertical coordinates of each point are given by y, solve each equation for y and plug in 6 for x, as follows:
Taking the difference of the resulting y -values give the vertical distance between the points (6,27) and (6,48), which is 21.
Since the vertical coordinates of each point are given by y, solve each equation for y and plug in 6 for x, as follows:
Taking the difference of the resulting y -values give the vertical distance between the points (6,27) and (6,48), which is 21.
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Solve the following system of equations:
–2x + 3y = 10
2x + 5y = 6
Solve the following system of equations:
–2x + 3y = 10
2x + 5y = 6
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Since we have –2x and +2x in the equations, it makes sense to add the equations together to give 8y = 16 yielding y = 2. Then we substitute y = 2 into one of the original equations to get x = –2. So the solution to the system of equations is (–2, 2)
Since we have –2x and +2x in the equations, it makes sense to add the equations together to give 8y = 16 yielding y = 2. Then we substitute y = 2 into one of the original equations to get x = –2. So the solution to the system of equations is (–2, 2)
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Find the point where the line y = .25(x – 20) + 12 crosses the x-axis.
Find the point where the line y = .25(x – 20) + 12 crosses the x-axis.
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When the line crosses the x-axis, the y-coordinate is 0. Substitute 0 into the equation for y and solve for x.
.25(x – 20) + 12 = 0
.25_x_ – 5 = –12
.25_x_ = –7
x = –28
The answer is the point (–28,0).
When the line crosses the x-axis, the y-coordinate is 0. Substitute 0 into the equation for y and solve for x.
.25(x – 20) + 12 = 0
.25_x_ – 5 = –12
.25_x_ = –7
x = –28
The answer is the point (–28,0).
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What is the slope of the line perpendicular to the line given by the equation
6x – 9y +14 = 0
What is the slope of the line perpendicular to the line given by the equation
6x – 9y +14 = 0
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First rearrange the equation so that it is in slope-intercept form, resulting in y=2/3 x + 14/9. The slope of this line is 2/3, so the slope of the line perpendicular will have the opposite reciprocal as a slope, which is -3/2.
First rearrange the equation so that it is in slope-intercept form, resulting in y=2/3 x + 14/9. The slope of this line is 2/3, so the slope of the line perpendicular will have the opposite reciprocal as a slope, which is -3/2.
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What is the slope of the line perpendicular to the line represented by the equation y = -2x+3?
What is the slope of the line perpendicular to the line represented by the equation y = -2x+3?
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Perpendicular lines have slopes that are the opposite of the reciprocal of each other. In this case, the slope of the first line is -2. The reciprocal of -2 is -1/2, so the opposite of the reciprocal is therefore 1/2.
Perpendicular lines have slopes that are the opposite of the reciprocal of each other. In this case, the slope of the first line is -2. The reciprocal of -2 is -1/2, so the opposite of the reciprocal is therefore 1/2.
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What is the distance between the two points, (1,1) and (7,9)?
What is the distance between the two points, (1,1) and (7,9)?
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distance2 = (_x_2 – _x_1)2 + (_y_2 – _y_1)2
Looking at the two order pairs given, _x_1 = 1, _y_1 = 1, _x_2 = 7, _y_2 = 9.
distance2 = (7 – 1)2 + (9 – 1)2 = 62 + 82 = 100
distance = 10
distance2 = (_x_2 – _x_1)2 + (_y_2 – _y_1)2
Looking at the two order pairs given, _x_1 = 1, _y_1 = 1, _x_2 = 7, _y_2 = 9.
distance2 = (7 – 1)2 + (9 – 1)2 = 62 + 82 = 100
distance = 10
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