Perpendicular Lines - PSAT Math
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Two points on line m are (3,7) and (-2, 5). Line k is perpendicular to line m. What is the slope of line k?
Two points on line m are (3,7) and (-2, 5). Line k is perpendicular to line m. What is the slope of line k?
The slope of line m is the (y2 - y1) / (x2 - x1) = (5-7) / (-2 - 3)
= -2 / -5
= 2/5
To find the slope of a line perpendicular to a given line, we must take the negative reciprocal of the slope of the given line.
Thus the slope of line k is the negative reciprocal of 2/5 (slope of line m), which is -5/2.
The slope of line m is the (y2 - y1) / (x2 - x1) = (5-7) / (-2 - 3)
= -2 / -5
= 2/5
To find the slope of a line perpendicular to a given line, we must take the negative reciprocal of the slope of the given line.
Thus the slope of line k is the negative reciprocal of 2/5 (slope of line m), which is -5/2.
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The equation of a line is: 8x + 16y = 48
What is the slope of a line that runs perpendicular to that line?
The equation of a line is: 8x + 16y = 48
What is the slope of a line that runs perpendicular to that line?
First, solve for the equation of the line in the form of y = mx + b so that you can determine the slope, m of the line:
8x + 16y = 48
16y = -8x + 48
y = -(8/16)x + 48/16
y = -(1/2)x + 3
Therefore the slope (or m) = -1/2
The slope of a perpendicular line is the negative inverse of the slope.
m = - (-2/1) = 2
First, solve for the equation of the line in the form of y = mx + b so that you can determine the slope, m of the line:
8x + 16y = 48
16y = -8x + 48
y = -(8/16)x + 48/16
y = -(1/2)x + 3
Therefore the slope (or m) = -1/2
The slope of a perpendicular line is the negative inverse of the slope.
m = - (-2/1) = 2
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A line passes through (2, 8) and (4, 15). What is a possible equation for a line perpendicular to this one?
A line passes through (2, 8) and (4, 15). What is a possible equation for a line perpendicular to this one?
Remember, perpendicular lines have opposite-reciprocal slopes; therefore, let's first find the slope of our line. That is found by the equation: rise/run or y2 – y1/x2 – x1
Substituting in our values: (15 – 8)/(4 – 2) = 7/2
The perpendicular slope is therefore –2/7.
Since ANY perpendicular line will intersect with this line at some point. We merely need to choose the answer that has a line with slope –2/7. Following the slope intercept form (y = mx + b), we know that the coefficient of x will give us this; therefore our answer is: y = (–2/7)x + 4
Remember, perpendicular lines have opposite-reciprocal slopes; therefore, let's first find the slope of our line. That is found by the equation: rise/run or y2 – y1/x2 – x1
Substituting in our values: (15 – 8)/(4 – 2) = 7/2
The perpendicular slope is therefore –2/7.
Since ANY perpendicular line will intersect with this line at some point. We merely need to choose the answer that has a line with slope –2/7. Following the slope intercept form (y = mx + b), we know that the coefficient of x will give us this; therefore our answer is: y = (–2/7)x + 4
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What line is perpendicular to x + 3_y_ = 6 and travels through point (1,5)?
What line is perpendicular to x + 3_y_ = 6 and travels through point (1,5)?
Convert the equation to slope intercept form to get y = –1/3_x_ + 2. The old slope is –1/3 and the new slope is 3. Perpendicular slopes must be opposite reciprocals of each other: _m_1 * _m_2 = –1
With the new slope, use the slope intercept form and the point to calculate the intercept: y = mx + b or 5 = 3(1) + b, so b = 2
So y = 3_x_ + 2
Convert the equation to slope intercept form to get y = –1/3_x_ + 2. The old slope is –1/3 and the new slope is 3. Perpendicular slopes must be opposite reciprocals of each other: _m_1 * _m_2 = –1
With the new slope, use the slope intercept form and the point to calculate the intercept: y = mx + b or 5 = 3(1) + b, so b = 2
So y = 3_x_ + 2
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Which set of lines is perpendicular?
Which set of lines is perpendicular?
Two lines are perpendicular to each other if their slopes are negative reciprocals. For example, if one line has a slope of 2, the line perpendicular to it has a slope of –1/2. One easy way to eliminate answer choices is to check if the slopes have the same sign, i.e. both positive or both negative. If so, they cannot be perpendicular. Several of the lines in the answer choices are of the form y = mx + b, where m is the slope and b is the y-intercept. We are only worried about the slope for the purposes of this question.
y = 3_x_ + 5 and y = 5_x_ + 3 both have positive slopes (m = 3 and m = 5, respectively), so they aren't perpendicular.
y = 3_x_/5 – 3 and y = 5_x_/3 + 3 both have positive slopes, so again they aren't perpendicular.
y = x – 1/2 and y = –x + 1/2 have slopes of m = 1 and m = –1, respectively. One is positive and one is negative, so that is a good sign. Let's take the negative reciprocal of 1. 1 /–1 = –1. So these two slopes are in fact negative reciprocals, and these two lines are perpendicular to each other. Even though we have found the correct answer, let's go through the other two choices to be sure.
The line between the points (1,3) and (3,5), and y = 4_x_ + 7: We need to find the slope of the first line. slope = rise / run = (_y_2 – _y_1) / (_x_2 – x_1) = (5 – 3) / (3 – 1) = 1. The slope of y = 4_x + 7 is also positive (m = 4), so the lines are not perpendicular.
The line between the points (7,4) and (4,7), and the line between the points (3,9) and (4,8): the first slope = (7 – 4) / (4 – 7) = –1 and the second slope = (8 – 9) / (4 – 3) = –1. They have the same slope, making them parallel, not perpendicular.
Two lines are perpendicular to each other if their slopes are negative reciprocals. For example, if one line has a slope of 2, the line perpendicular to it has a slope of –1/2. One easy way to eliminate answer choices is to check if the slopes have the same sign, i.e. both positive or both negative. If so, they cannot be perpendicular. Several of the lines in the answer choices are of the form y = mx + b, where m is the slope and b is the y-intercept. We are only worried about the slope for the purposes of this question.
y = 3_x_ + 5 and y = 5_x_ + 3 both have positive slopes (m = 3 and m = 5, respectively), so they aren't perpendicular.
y = 3_x_/5 – 3 and y = 5_x_/3 + 3 both have positive slopes, so again they aren't perpendicular.
y = x – 1/2 and y = –x + 1/2 have slopes of m = 1 and m = –1, respectively. One is positive and one is negative, so that is a good sign. Let's take the negative reciprocal of 1. 1 /–1 = –1. So these two slopes are in fact negative reciprocals, and these two lines are perpendicular to each other. Even though we have found the correct answer, let's go through the other two choices to be sure.
The line between the points (1,3) and (3,5), and y = 4_x_ + 7: We need to find the slope of the first line. slope = rise / run = (_y_2 – _y_1) / (_x_2 – x_1) = (5 – 3) / (3 – 1) = 1. The slope of y = 4_x + 7 is also positive (m = 4), so the lines are not perpendicular.
The line between the points (7,4) and (4,7), and the line between the points (3,9) and (4,8): the first slope = (7 – 4) / (4 – 7) = –1 and the second slope = (8 – 9) / (4 – 3) = –1. They have the same slope, making them parallel, not perpendicular.
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If two lines have slopes of 
and 
, which statement about the lines is true?
If two lines have slopes of
Perpendicular lines have slopes that are the negative reciprocals of each other.
Perpendicular lines have slopes that are the negative reciprocals of each other.
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Which of the following lines is perpendicular to 
Which of the following lines is perpendicular to
The line which is perpendicular has a slope which is the negative inverse of the slope of the original line.
The line which is perpendicular has a slope which is the negative inverse of the slope of the original line.
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Which set of lines is perpendicular?
Which set of lines is perpendicular?
Two lines are perpendicular to each other if their slopes are negative reciprocals. For example, if one line has a slope of 2, the line perpendicular to it has a slope of –1/2. One easy way to eliminate answer choices is to check if the slopes have the same sign, i.e. both positive or both negative. If so, they cannot be perpendicular. Several of the lines in the answer choices are of the form y = mx + b, where m is the slope and b is the y-intercept. We are only worried about the slope for the purposes of this question.
y = 3_x_ + 5 and y = 5_x_ + 3 both have positive slopes (m = 3 and m = 5, respectively), so they aren't perpendicular.
y = 3_x_/5 – 3 and y = 5_x_/3 + 3 both have positive slopes, so again they aren't perpendicular.
y = x – 1/2 and y = –x + 1/2 have slopes of m = 1 and m = –1, respectively. One is positive and one is negative, so that is a good sign. Let's take the negative reciprocal of 1. 1 /–1 = –1. So these two slopes are in fact negative reciprocals, and these two lines are perpendicular to each other. Even though we have found the correct answer, let's go through the other two choices to be sure.
The line between the points (1,3) and (3,5), and y = 4_x_ + 7: We need to find the slope of the first line. slope = rise / run = (_y_2 – _y_1) / (_x_2 – x_1) = (5 – 3) / (3 – 1) = 1. The slope of y = 4_x + 7 is also positive (m = 4), so the lines are not perpendicular.
The line between the points (7,4) and (4,7), and the line between the points (3,9) and (4,8): the first slope = (7 – 4) / (4 – 7) = –1 and the second slope = (8 – 9) / (4 – 3) = –1. They have the same slope, making them parallel, not perpendicular.
Two lines are perpendicular to each other if their slopes are negative reciprocals. For example, if one line has a slope of 2, the line perpendicular to it has a slope of –1/2. One easy way to eliminate answer choices is to check if the slopes have the same sign, i.e. both positive or both negative. If so, they cannot be perpendicular. Several of the lines in the answer choices are of the form y = mx + b, where m is the slope and b is the y-intercept. We are only worried about the slope for the purposes of this question.
y = 3_x_ + 5 and y = 5_x_ + 3 both have positive slopes (m = 3 and m = 5, respectively), so they aren't perpendicular.
y = 3_x_/5 – 3 and y = 5_x_/3 + 3 both have positive slopes, so again they aren't perpendicular.
y = x – 1/2 and y = –x + 1/2 have slopes of m = 1 and m = –1, respectively. One is positive and one is negative, so that is a good sign. Let's take the negative reciprocal of 1. 1 /–1 = –1. So these two slopes are in fact negative reciprocals, and these two lines are perpendicular to each other. Even though we have found the correct answer, let's go through the other two choices to be sure.
The line between the points (1,3) and (3,5), and y = 4_x_ + 7: We need to find the slope of the first line. slope = rise / run = (_y_2 – _y_1) / (_x_2 – x_1) = (5 – 3) / (3 – 1) = 1. The slope of y = 4_x + 7 is also positive (m = 4), so the lines are not perpendicular.
The line between the points (7,4) and (4,7), and the line between the points (3,9) and (4,8): the first slope = (7 – 4) / (4 – 7) = –1 and the second slope = (8 – 9) / (4 – 3) = –1. They have the same slope, making them parallel, not perpendicular.
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Line 
includes the points 
and 
. Line 
includes the points 
and 
. Which of the following statements is true of these lines?
Line
We calculate the slopes of the lines using the slope formula.
The slope of line 
is
The slope of line 
is
Parallel lines and identical lines must have the same slope, so these can be eliminated as choices. The slopes of perpendicular lines must have product 
. The slopes have product
so they are not perpendicular.
The correct response is that the lines are distinct but neither parallel nor perpendicular.
We calculate the slopes of the lines using the slope formula.
The slope of line
The slope of line
Parallel lines and identical lines must have the same slope, so these can be eliminated as choices. The slopes of perpendicular lines must have product
so they are not perpendicular.
The correct response is that the lines are distinct but neither parallel nor perpendicular.
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Line 
includes the points 
and 
. Line 
includes the points 
and 
. Which of the following statements is true of these lines?
Line
We calculate the slopes of the lines using the slope formula.
The slope of line 
is
The slope of line 
is
The slopes are not the same, so the lines are neither parallel nor identical. We multiply their slopes to test for perpendicularity:
The product of the slopes is 
, making the lines perpendicular.
We calculate the slopes of the lines using the slope formula.
The slope of line
The slope of line
The slopes are not the same, so the lines are neither parallel nor identical. We multiply their slopes to test for perpendicularity:
The product of the slopes is
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Consider the equations 
and 
. Which of the following statements is true of the lines of these equations?
Consider the equations
We find the slope of each line by putting each equation in slope-intercept form 
and examining the coefficient of 
.
is already in slope-intercept form; its slope is 
.
To get 
in slope-intercept form we solve for 
:
The slope of this line is also .
The slopes are equal; however, the 
-intercepts are different - the 
-intercept of the first line is 
and that of the second line is 
. Therefore, the lines are parallel as opposed to being the same line.
We find the slope of each line by putting each equation in slope-intercept form
To get
The slope of this line is also
The slopes are equal; however, the
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Which line is perpendicular to 
?
Which line is perpendicular to
To determine if lines are perpendicular, look at the slope. Perpendicular lines have slopes that are negative reciprocals. The slope of 
is 
. The negative reciprocal of 
is 
. Therefore, the answer is 
.
To determine if lines are perpendicular, look at the slope. Perpendicular lines have slopes that are negative reciprocals. The slope of
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What is the equation of a line that runs perpendicular to the line 2_x_ + y = 5 and passes through the point (2,7)?
What is the equation of a line that runs perpendicular to the line 2_x_ + y = 5 and passes through the point (2,7)?
First, put the equation of the line given into slope-intercept form by solving for y. You get y = -2_x_ +5, so the slope is –2. Perpendicular lines have opposite-reciprocal slopes, so the slope of the line we want to find is 1/2. Plugging in the point given into the equation y = 1/2_x_ + b and solving for b, we get b = 6. Thus, the equation of the line is y = ½_x_ + 6. Rearranged, it is –x/2 + y = 6.
First, put the equation of the line given into slope-intercept form by solving for y. You get y = -2_x_ +5, so the slope is –2. Perpendicular lines have opposite-reciprocal slopes, so the slope of the line we want to find is 1/2. Plugging in the point given into the equation y = 1/2_x_ + b and solving for b, we get b = 6. Thus, the equation of the line is y = ½_x_ + 6. Rearranged, it is –x/2 + y = 6.
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The equation of line p is y= 1/4x +6. If line k contains the point (3,5) and is perpendicular to line p, find the equation of line k.
The equation of line p is y= 1/4x +6. If line k contains the point (3,5) and is perpendicular to line p, find the equation of line k.
Using the slope intercept formula, we can see the slope of line p is ¼. Since line k is perpendicular to line p it must have a slope that is the negative reciprocal. (-4/1) If we set up the formula y=mx+b, using the given point and a slope of (-4), we can solve for our b or y-intercept. In this case it would be 17.
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Using the slope intercept formula, we can see the slope of line p is ¼. Since line k is perpendicular to line p it must have a slope that is the negative reciprocal. (-4/1) If we set up the formula y=mx+b, using the given point and a slope of (-4), we can solve for our b or y-intercept. In this case it would be 17.
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Solve the system of equations for the point of intersection.
Solve the system of equations for the point of intersection.
First one needs to use one of the two equations to substitute one of the unknowns.
From the second equation we can derive that y = x – 3.
Then we substitute what we got into the first equation which gives us: x + x – 3 = 15.
Next we solve for x, so 2_x_ = 18 and x = 9.
x – y = 3, so y = 6.
These two lines will intersect at the point (9,6).
First one needs to use one of the two equations to substitute one of the unknowns.
From the second equation we can derive that y = x – 3.
Then we substitute what we got into the first equation which gives us: x + x – 3 = 15.
Next we solve for x, so 2_x_ = 18 and x = 9.
x – y = 3, so y = 6.
These two lines will intersect at the point (9,6).
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In the xy-plane, the equation of the line n is –x+8y=17. If the line m is perpendicular to line n, what is a possible equation of line m?
In the xy-plane, the equation of the line n is –x+8y=17. If the line m is perpendicular to line n, what is a possible equation of line m?
We start by add x to the other side of the equation to get the y by itself, giving us 8y =17 + x. We then divide both sides by 8, giving us y= 17/8 + 1/8x. Since we are looking for the equation of a perpendicular line, we know the slope (the coefficient in front of x) will be the opposite reciprocal of the slope of our line, giving us y= -8x + 5 as the answer.
We start by add x to the other side of the equation to get the y by itself, giving us 8y =17 + x. We then divide both sides by 8, giving us y= 17/8 + 1/8x. Since we are looking for the equation of a perpendicular line, we know the slope (the coefficient in front of x) will be the opposite reciprocal of the slope of our line, giving us y= -8x + 5 as the answer.
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Which of the following equations represents a line that goes through the point 
and is perpendicular to the line 
?
Which of the following equations represents a line that goes through the point 
In order to solve this problem, we need first to transform the equation from standard form to slope-intercept form:
Transform the original equation to find its slope.
First, subtract 
from both sides of the equation.
Simplify and rearrange.
Next, divide both sides of the equation by 6.
The slope of our first line is equal to 
. Perpendicular lines have slopes that are opposite reciprocals of each other; therefore, if the slope of one is x, then the slope of the other is equal to the following:
Let's calculate the opposite reciprocal of our slope:
The slope of our line is equal to 2. We now have the following partial equation:
We are missing the y-intercept, 
. Substitute the x- and y-values in the given point 
to solve for the missing y-intercept.
Add 4 to both sides of the equation.
Substitute this value into our partial equation to construct the equation of our line:
In order to solve this problem, we need first to transform the equation from standard form to slope-intercept form:
Transform the original equation to find its slope.
First, subtract 
Simplify and rearrange.
Next, divide both sides of the equation by 6.
The slope of our first line is equal to 
Let's calculate the opposite reciprocal of our slope:
The slope of our line is equal to 2. We now have the following partial equation:
We are missing the y-intercept, 
Add 4 to both sides of the equation.
Substitute this value into our partial equation to construct the equation of our line:
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Line m passes through the points (1, 4) and (5, 2). If line p is perpendicular to m, then which of the following could represent the equation for p?
Line m passes through the points (1, 4) and (5, 2). If line p is perpendicular to m, then which of the following could represent the equation for p?
The slope of m is equal to y2-y1/x2-x1 = 2-4/5-1 = -1/2
Since line p is perpendicular to line m, this means that the products of the slopes of p and m must be **–**1:
(slope of p) * (-1/2) = -1
Slope of p = 2
So we must choose the equation that has a slope of 2. If we rewrite the equations in point-slope form (y = mx + b), we see that the equation 2x – y = 3 could be written as y = 2x – 3. This means that the slope of the line 2x – y =3 would be 2, so it could be the equation of line p. The answer is 2x – y = 3.
The slope of m is equal to y2-y1/x2-x1 = 2-4/5-1 = -1/2
Since line p is perpendicular to line m, this means that the products of the slopes of p and m must be **–**1:
(slope of p) * (-1/2) = -1
Slope of p = 2
So we must choose the equation that has a slope of 2. If we rewrite the equations in point-slope form (y = mx + b), we see that the equation 2x – y = 3 could be written as y = 2x – 3. This means that the slope of the line 2x – y =3 would be 2, so it could be the equation of line p. The answer is 2x – y = 3.
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In the xy-coordinate plane, a line A contains the points (0,0) and (3,1). If the line B is perpendicular to A at (3,1), what is the equation of the line?
In the xy-coordinate plane, a line A contains the points (0,0) and (3,1). If the line B is perpendicular to A at (3,1), what is the equation of the line?
First, you need to obtain the equation of the first line, A. Its slope is given by:
(y2 - y1) / (x2 - x1) = (1 - 0) / (3 - 0) = 1/3 = slope of A.
Remember that the slope of a perpendicular line to a given line is -1 times the inverse of its slope. Thus the slope of B:
(-1) x 1 / (1/3) = -3
Thus with y = mx + b, m = -3. Now the line must include (3,1). Thus:
with y = -3x + b:
1 = -3(3) + b;
1 = -9 + b; add 9 to both sides:
10 = b
First, you need to obtain the equation of the first line, A. Its slope is given by:
(y2 - y1) / (x2 - x1) = (1 - 0) / (3 - 0) = 1/3 = slope of A.
Remember that the slope of a perpendicular line to a given line is -1 times the inverse of its slope. Thus the slope of B:
(-1) x 1 / (1/3) = -3
Thus with y = mx + b, m = -3. Now the line must include (3,1). Thus:
with y = -3x + b:
1 = -3(3) + b;
1 = -9 + b; add 9 to both sides:
10 = b
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What line is perpendicular to the line 2x + 3y = 6 through (4, 1)?
What line is perpendicular to the line 2x + 3y = 6 through (4, 1)?
The given equation is in standard form, so it must be converted to slope-intercept form: y = mx + b to discover the slope is –2/3. To be perpendicular the new slope must be 3/2 (opposite reciprocal of the old slope). Using the new slope and the given point we can substitute these values back into the slope-intercept form to find the new intercept, –5. In slope-intercept form the new equation is y = 3/2x – 5. The correct answer is this equation converted to standard form.
The given equation is in standard form, so it must be converted to slope-intercept form: y = mx + b to discover the slope is –2/3. To be perpendicular the new slope must be 3/2 (opposite reciprocal of the old slope). Using the new slope and the given point we can substitute these values back into the slope-intercept form to find the new intercept, –5. In slope-intercept form the new equation is y = 3/2x – 5. The correct answer is this equation converted to standard form.
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