Perpendicular Lines - GMAT Quantitative Reasoning

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Question

What is the equation of the line that is perpendicular to and goes through point ?

Answer

Perpendicular lines have slopes that are negative reciprocals of each other.

The slope for the given line is , from , where is the slope. Therefore, the negative reciprocal is $-\frac{1}{2}$ .

$m=-\frac{1}{2}$ and :

$y-y_{1}=m(x-x_{1})$

$y-1=-\frac{1}{2}(x-5)$

$y-1=-\frac{1}{2}x+\frac{5}{2}$

$y=-\frac{1}{2}x+\frac{5}{2}+1$

$y=-\frac{1}{2}x+\frac{5}{2}+\frac{2}{2}$

$y=-\frac{1}{2}x+\frac{7}{2}$

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Question

Write the equation of a line that is perpendicular to $y=\frac{-1}{2}x+4$ and goes through point ?

Answer

A perpendicular line has a negative reciprocal slope to the given line.

The given line, $y=\frac{-1}{2}x+4$ , has a slope of $-\frac{1}{2}$ , as is the slope in the standard form equation .

Slope of perpendicular line:

Point:

Using the point slope formula, we can solve for the equation:

$y-y_{1}=m(x-x_{1})$

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Question

Determine the equation of a line perpendicular to at the point .

Answer

The equation of a line in standard form is written as follows:

Where is the slope of the line and is the y intercept. First, we can determine the slope of the perpendicular line using the knowledge that its slope must be the negative reciprocal of the slope of the line to which it is perpendicular. For the given line, we can see that , so the slope of a line perpendicular to it will be the negative reciprocal of that value, which gives us:

$m_{perp}=-\frac{1}{m}=-\frac{1}{3}$

Now that we know the slope of the perpendicular line, we can plug its value into the formula for a line along with the coordinates of the given point, allowing us to calculate the -intercept, :

$4=-\frac{1}{3}(2)+b$

$b=\frac{14}{3}$

We now have the slope and the -intercept of the perpendicular line, which is all we need to write its equation in standard form:

$y=-\frac{1}{3}x+\frac{14}{3}$

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Question

Given , find the equation of a line that is perpendicular to and goes through the point .

$\small h(x)=-\frac{5}3x-12$

Answer

Given

$\small h(x)=-\frac{5}3x-12$

We need a perpendicular line going through (14,0).

Perpendicular lines have opposite reciprocal slopes.

So we get our slope to be

$\small \frac{3}{5}$

Next, plug in all our knowns into and solve for .

$0=14\cdot \frac{3}{5}+b$

$\small b=-\frac{42}{5}$ .

Making our answer

$\small y=\frac{3x}{5}-\frac{42}{5}$ .

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Question

Given the function , which of the following is the equation of a line perpendicular to and has a -intercept of ?

Answer

Given a line defined by the equation with slope , any line that is perpendicular to must have a slope $-\frac{1}{m}$ , or the negative reciprocal of .

Since , the slope is and the slope of any line parallel to must have a slope of $-\frac{1}{m}=-\frac{1}{4}$ .

Since also needs to have a -intercept of , then the equation for must be $g(x)=-\frac{1}{4}x+7$ .

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Question

Given the function , which of the following is the equation of a line perpendicular to and has a -intercept of ?

Answer

Given a line defined by the equation with slope , any line that is perpendicular to must have a slope $-\frac{1}{m}$ , or the negative reciprocal of .

Since , the slope is and the slope of any line parallel to must have a slope of $-\frac{1}{m}=-\frac{1}{-5}=\frac{1}{5}$ .

Since also needs to have a -intercept of , then the equation for must be $g(x)=\frac{1}{5}x+8$ .

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Question

Given the function , which of the following is the equation of a line perpendicular to and has a -intercept of ?

Answer

Given a line defined by the equation with slope , any line that is perpendicular to must have a slope $-\frac{1}{m}$ , or the negative reciprocal of .

Since , the slope is and the slope of any line parallel to must have a slope of $-\frac{1}{m}=-\frac{1}{7}$ .

Since also needs to have a -intercept of , then the equation for must be $g(x)=-\frac{1}{7}x-9$ .

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Question

What is the slope of the line perpendicular to ?

Answer

Perpendicular lines have slopes that are negative reciprocals of each other. Therefore, rewrite the equation in slope intercept form :

Slope of given line:

Negative reciprocal:

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Question

What is the slope of a line perpendicular to the line of the equation ?

Answer

The graph of for any real number is a horizontal line. A line parallel to it is a vertical line, which has a slope that is undefined.

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Question

Line 1 is the line of the equation . Line 2 is perpendicular to this line. What is the slope of Line 2?

Answer

Rewrite in slope-intercept form:

The slope of the line is the coefficient of , which is . A line perpendicular to this has as its slope the opposite of the reciprocal of :

$m = -\frac{1}{-1} = 1$

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Question

Given:

$\small f(x)=\frac{5}{6}x-17$

Calculate the slope of , a line perpendicular to .

Answer

To find the slope of a line perpendicular to a given line, simply take the opposite reciprocal of the slope of the given line.

Since f(x) is given in slope intercept form,

$y=mx+b \rightarrow m=slope$ .

Therefore our original slope is

$m=\small \frac{5}{6}$

So our new slope becomes:

$m_{new}=-\frac{1}{m}=-\frac{1}{\frac{5}{6}}=\small -\frac{6}{5}$

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Question

What would be the slope of a line perpendicular to the following line?

Answer

The equation for a line in standard form is written as follows:

Where is the slope of the line and is the y intercept. By definition, the slope of a line is the negative reciprocal of the slope of the line to which it is perpendicular. So if the given line has a slope of , the slope of any line perpendicular to it will have the negative reciprocal of that slope. This gives us:

$y=-5x-3\rightarrow m=-5$

$m_{perp}=-\frac{1}{m}=-\frac{1}{(-5)}=\frac{1}{5}$

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Question

Give the slope of a line on the coordinate plane.

Statement 1: The line shares an -intercept and its -intercept with the line of the equation .

Statement 2: The line is perpendicular to the line of the equation .

Answer

Assume Statement 1 alone. In order to determine the slope of a line on the coordinate plane, the coordinates of two of its points are needed. The -intercept of the line of the equation can be found by substituting and solving for :

The -intercept of the line is at the origin, . It follows that the -intercept is also at the origin. Therefore, Statement 1 only gives one point on the line, and its slope cannot be determined.

Assume Statement 2 alone. The slope of the line of the equation can be calculated by putting it in slope-intercept form :

$7y\div 7 = -5x \div 7$

$y = -\frac{5}{7} x$

The slope of this line is the coefficient of , which is $-\frac{5}{7}$ . A line perpendicular to this one has as its slope the opposite of the reciprocal of $-\frac{5}{7}$ , which is

$- \frac{1}{-\frac{5}{7}} = \frac{7}{5}$ .

The question is answered.

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Question

Find the equation of the line that is perpendicular to the line connecting the points $\dpi{100} \small (0,-4)\ and\ (-1,-7)$ .

Answer

Lines are perpendicular if their slopes are negative reciprocals of each other. First we need to find the slope of the line in the question stem.

$slope = \frac{rise}{run} = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{-7 + 4}{-1 - 0} = \frac{-3}{-1} = 3$

The negative reciprocal of 3 is $\dpi{100} \small -\frac{1}{3}$ , so our answer will have a slope of $\dpi{100} \small -\frac{1}{3}$ . Let's go through the answer choices and see.

$\dpi{100} \small y=3x-1$ : This line is of the form $\dpi{100} \small y=mx+b$ , where $\dpi{100} \small m$ is the slope. The slope is 3, so this line is parallel, not perpendicular, to our line in question.

$\dpi{100} \small y=-4x+8$ : The slope here is $\dpi{100} \small -4$ , also wrong.

$\dpi{100} \small y=\frac{x}{3}+1$ : The slope of this line is $\dpi{100} \small \frac{1}{3}$ . This is the reciprocal, but not the negative reciprocal, so this is also incorrect.

The line between the points $\dpi{100} \small (3,0)\ and\ (-3,2)$ : $\dpi{100} \small slope = \frac{2}{(-3-3)}=\frac{2}{-6}=-\frac{1}{3}$ .

This is the correct answer! Let's check the last answer choice as well.

The line between points $\dpi{100} \small (0,0)\ and\ (2,2)$ : $\dpi{100} \small slope = \frac{2}{2}=1$ , which is incorrect.

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Question

Determine whether the lines with equations and are perpendicular.

Answer

If two equations are perpendicular, then they will have inverse negative slopes of each other. So if we compare the slopes of the two equations, then we can find the answer. For the first equation we have $3y+x=2\Rightarrow y=\frac{-1}{3}x+\frac{2}{3}$

so the slope is $-\frac{1}{3}$ .

So for the equations to be perpendicular, the other equation needs to have a slope of 3. For the second equation, we have

$4y-2x=3\Rightarrow y=\frac{1}{2}x+\frac{3}{4}$

so the slope is $\frac{1}{2}$ .

Since the slope of the second equation is not equal to 3, then the lines are not perpendicular.

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Question

Figure NOT drawn to scale.

Refer to the above figure.

True or false: $m \perp t$

Statement 1: $\angle 4$ is a right angle.

Statement 2: $\angle 3$ and $\angle 5$ are supplementary.

Answer

Statement 1 alone establishes by definition that $l\perp t$ , but does not establish any relationship between and .

By Statement 2 alone, since same-side interior angles are supplementary, , but no conclusion can be drawn about the relationship of , since the actual measures of the angles are not given.

Assume both statements are true. If two lines are parallel, then any line in their plane perpendicular to one must be perpendicular to the other. and $l\perp t$ , so it can be established that $m \perp t$ .

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Question

Refer to the above figure. . True or false: $m \perp t$

Statement 1: $\angle 3 \cong \angle 5$

Statement 2: $\angle 2$ and $\angle 6$ are supplementary.

Answer

If transversal crosses two parallel lines and , then same-side interior angles are supplementary, so $\angle 3$ and $\angle 5$ are supplementary angles. Also, corresponding angles are congruent, so $\angle 2 \cong \angle 6$ .

By Statement 1 alone, angles $\angle 3$ and $\angle 5$ are congruent as well as supplementary; by Statement 2 alone, $\angle 2$ and $\angle 6$ are also supplementary as well as congruent. Two angles that are both supplementary and congruent are both right angles, so from either statement alone, and intersect at right angles, so, consequently, $m \perp t$ .

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Question

Find the equation of the line that is perpendicular to the following equation and passes through the point .

$y=-\frac{x}{2}+14$

Answer

To solve this equation, we want to begin by recalling how to find the slope of a perpendicular line. In this case, our original line is modeled by the following:

$y=-\frac{x}{2}+14$

To find the slope of any line perpendicular to the above equation, we simply need to take the reciprocal of the first slope, and then change its sign. Our original slope is $-\frac{1}{2}$ , so

$m=-\frac{1}{2}$

becomes

.

If we flip $\frac{1}{2}$ , we get , and the opposite sign of a negative is a positive; hence, our slope is positive .

So, we know our perpendicular line should look something like this:

However, we need to find out what (our -intercept) is in order to complete our equation. To do so, we need to plug in the ordered pair we received in the question, , and solve for :

So, by putting everything together, we get our final equation:

This equation satisfies the conditions of being perpendicular to our initial equation and passing through .

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Question

Which of the following lines is perpendicular to $y=-\frac{3}{4}x+7$ ?

Answer

In order for a line to be perpendicular to another line defined by the equation , the slope of line must be a negative reciprocal of the slope of line . Since line 's slope is in the slope-intercept equation above, line 's slope would therefore be $-\frac{1}{m}$ .

In this instance, $m=-\frac{3}{4}$ , so $-\frac{1}{m}=-\frac{1}{-\frac{3}{4}}=\frac{4}{3}$ . Therefore, the correct solution is $y=\frac{4}{3}x+7$ .

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Question

A given line has a slope of . What is the slope of any line perpendicular to ?

Answer

In order for a line to be perpendicular to another line defined by the equation , the slope of line must be a negative reciprocal of the slope of line . Since line 's slope is in the slope-intercept equation above, line 's slope would therefore be $-\frac{1}{m}$ .

Given that we have a line with a slope , we can therefore conclude that any perpendicular line would have a slope $-\frac{1}{m}=-\frac{1}{6}$ .

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