Calculating whether lines are perpendicular - GMAT Quantitative

Card 0 of 176

Two perpendicular lines intersect at point . One line passes through ; the other, through . What is the value of ?

Tap to see back →

The slope of the first line, in terms of , is

The slope of the second line is

The slopes of two perpendicular lines have product , so we set up this equation and solve for :

$$\frac{Y-7}{2}$ \cdot $\frac{Y-4}{-5}$ = -1$

$$\frac{ \left( Y-7 \right) \left( Y-4 \right) }{-10}$ = -1$

$\left ( Y-7 \right ) \left ( Y-4 \right ) = 10$

$Y ^{2}-11Y + 28 = 10$

$Y ^{2}-11Y + 18 = 0$

$\left ( Y - 2 \right ) \left ( Y - 9 \right ) = 0$

$Y - 2 = 0 \Rightarrow Y = 2$

or

$Y - 9 = 0 \Rightarrow Y = 9$

Which of the following choices give the slopes of two perpendicular lines?

Tap to see back →

We can eliminate the choice immediately since the slopes of two perpendicular lines cannot have the same sign. We can also eliminate and undefined, , since a line with slope 0 and a line with undefined slope are perpendicular to each other, not a line of slope -1 or 1.

Of the two remaining choices, we check for the choice that includes two numbers whose product is -1.

$-1.75 \times 1.75 = 3.0625$ and $-0.8 \times 1.25 = -1$

so is the correct choice.

Which of the following is perpendicular to the line given by the equation:

$y=\frac{2}{3}$x-11$

Tap to see back →

In order for one line to be perpendicular to another, its slope must be the negative reciprocal of that line's slope. That is, the slope of any perpendicular line must be opposite in sign and the inverse of the slope of the line to which it is perpendicular:

In the given line we can see that $m=\frac{2}{3}$$ , so the slope of any line perpendicular to it will be:

There is only one answer choice with this slope, so we know the following line is perpendicular to the line given in the problem:

$y=-$\frac{3}{2}$x+7$

What is the slope of any line that is perpendicular to $y=-$\frac{1}{2}$x+7$ ?

Tap to see back →

For a given line defined by the equation with slope , any line perpendicular to has a slope of $-$\frac{1}{m}$$ , or the negative reciprocal of . Since the slope of the provided line in this instance is $m=-$\frac{1}{2}$$ , then the slope of any line perpendicular to is $-$\frac{1}{m}$=-$\frac{1}{-\frac{1}${2}}=2$ .

What is the slope of any line that is perpendicular to ?

Tap to see back →

For a given line defined by the equation with slope , any line perpendicular to has a slope of $-$\frac{1}{m}$$ , or the negative reciprocal of . Since the slope of the provided line in this instance is , then the slope of any line perpendicular to is $-$\frac{1}{m}$=-$\frac{1}{7}$$ .

What is the slope of any line that is perpendicular to $y=\frac{2}{3}$x+5$ ?

Tap to see back →

For a given line defined by the equation with slope , any line perpendicular to has a slope of $-$\frac{1}{m}$$ , or the negative reciprocal of . Since the slope of the provided line in this instance is $m=\frac{2}{3}$$ , then the slope of any line perpendicular to is $-$\frac{1}{m}$=-$\frac{1}{\frac{2}${3}}=-$\frac{3}{2}$$ .

Two perpendicular lines intersect at point . One line passes through ; the other, through . What is the value of ?

Tap to see back →

The slope of the first line, in terms of , is

The slope of the second line is

The slopes of two perpendicular lines have product , so we set up this equation and solve for :

$$\frac{Y-7}{2}$ \cdot $\frac{Y-4}{-5}$ = -1$

$$\frac{ \left( Y-7 \right) \left( Y-4 \right) }{-10}$ = -1$

$\left ( Y-7 \right ) \left ( Y-4 \right ) = 10$

$Y ^{2}-11Y + 28 = 10$

$Y ^{2}-11Y + 18 = 0$

$\left ( Y - 2 \right ) \left ( Y - 9 \right ) = 0$

$Y - 2 = 0 \Rightarrow Y = 2$

or

$Y - 9 = 0 \Rightarrow Y = 9$

Which of the following choices give the slopes of two perpendicular lines?

Tap to see back →

We can eliminate the choice immediately since the slopes of two perpendicular lines cannot have the same sign. We can also eliminate and undefined, , since a line with slope 0 and a line with undefined slope are perpendicular to each other, not a line of slope -1 or 1.

Of the two remaining choices, we check for the choice that includes two numbers whose product is -1.

$-1.75 \times 1.75 = 3.0625$ and $-0.8 \times 1.25 = -1$

so is the correct choice.

Which of the following is perpendicular to the line given by the equation:

$y=\frac{2}{3}$x-11$

Tap to see back →

In order for one line to be perpendicular to another, its slope must be the negative reciprocal of that line's slope. That is, the slope of any perpendicular line must be opposite in sign and the inverse of the slope of the line to which it is perpendicular:

In the given line we can see that $m=\frac{2}{3}$$ , so the slope of any line perpendicular to it will be:

There is only one answer choice with this slope, so we know the following line is perpendicular to the line given in the problem:

$y=-$\frac{3}{2}$x+7$

What is the slope of any line that is perpendicular to $y=-$\frac{1}{2}$x+7$ ?

Tap to see back →

For a given line defined by the equation with slope , any line perpendicular to has a slope of $-$\frac{1}{m}$$ , or the negative reciprocal of . Since the slope of the provided line in this instance is $m=-$\frac{1}{2}$$ , then the slope of any line perpendicular to is $-$\frac{1}{m}$=-$\frac{1}{-\frac{1}${2}}=2$ .

What is the slope of any line that is perpendicular to ?

Tap to see back →

For a given line defined by the equation with slope , any line perpendicular to has a slope of $-$\frac{1}{m}$$ , or the negative reciprocal of . Since the slope of the provided line in this instance is , then the slope of any line perpendicular to is $-$\frac{1}{m}$=-$\frac{1}{7}$$ .

What is the slope of any line that is perpendicular to $y=\frac{2}{3}$x+5$ ?

Tap to see back →

For a given line defined by the equation with slope , any line perpendicular to has a slope of $-$\frac{1}{m}$$ , or the negative reciprocal of . Since the slope of the provided line in this instance is $m=\frac{2}{3}$$ , then the slope of any line perpendicular to is $-$\frac{1}{m}$=-$\frac{1}{\frac{2}${3}}=-$\frac{3}{2}$$ .

Find the equation of the line that is perpendicular to the line connecting the points (0,-4) and (-1,-7).

Tap to see back →

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 -$\frac{1}{3}$, so our answer will have a slope of -$\frac{1}{3}$. Let's go through the answer choices and see.

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

y=-4x+8: The slope here is -4, also wrong.

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

The line between the points (3,0) and (-3,2):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 (0,0) and (2,2):slope = $\frac{2}{2}$=1, which is incorrect.

Determine whether the lines with equations and are perpendicular.

Tap to see back →

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.

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.

Tap to see back →

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 $m \perp t$ .

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.

Tap to see back →

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, $m \perp t$ .

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

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

Tap to see back →

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 .

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

Tap to see back →

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$ .

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

Tap to see back →

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}$$ .

Which of the following lines are perpendicular to $y=\frac{5}{6}$x+10$ ?

Tap to see back →

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}$$ .

Since in this instance the slope $m=\frac{5}{6}$$ , $-$\frac{1}{m}$=-$\frac{1}{\frac{5}${6}}=-$\frac{6}{5}$$ . Two of the above answers have this as their slope, so therefore that is the answer to our question.