Lines - GMAT Quantitative

Card 0 of 448

Question

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

Answer

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

$m_{1} =\frac{Y-7}{5-3} = \frac{Y-7}{2}$

The slope of the second line is

$m_{2} =\frac{Y-4}{5-10} = \frac{Y-4}{-5}$

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

$m_{1}m_{2} = 1$

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

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

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

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

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

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

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

or

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

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Question

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

Answer

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.

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Question

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

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

Answer

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:

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

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

$m_{perp}=-\frac{1}{m}=-\frac{1}{\frac{2}{3}}=-\frac{3}{2}$

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

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

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Question

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

Answer

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

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Question

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

Answer

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

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Question

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

Answer

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 $-\frac{1}{m}=-\frac{1}{\frac{2}{3}}=-\frac{3}{2}$ .

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Question

A right triangle is given with a missing value of . It is stated that the triangle is an acute right triangle with angles $50^\circ$ and . What is a possible value of in degrees?

Answer

It is important to recall that all triangles add to 180 degrees and a right triangle contains one angle that is equal to 90 degrees. Therefore, in this particular problem we can write the following equation to solve for the missing variable.

$\90^\circ+50^\circ+2t=180^\circ \140^\circ-2t=180^\circ \2t=180^\circ-140^\circ \2t=40^\circ \t=20$

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Question

What is the measure of an angle complementary to a $72 ^{\circ }$ angle?

Answer

Complementary angles have degree measures that total $90 ^{\circ }$ , so the measure of an angle complementary to a $72 ^{\circ }$ angle would have measure $(90-72)^{\circ } = 18 ^{\circ }$ .

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Question

What is the measure of an angle supplementary to a $66 ^{\circ }$ angle?

Answer

Supplementary angles have degree measures that total $180 ^{\circ }$ , so the measure of an angle complementary to a $66 ^{\circ }$ angle would have measure $\left ( 180- 66 \right )^{\circ } = 114^{\circ }$ .

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Question

What is the measure of an angle congruent to a $48 ^{\circ }$ angle?

Answer

Two angles are congruent if they have the same degree measure, so an angle will be congruent to a $48 ^{\circ }$ angle if its measure is also $48 ^{\circ }$ .

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Question

What is the measure of an angle that is supplementary to a $68^{\circ}$ angle?

Answer

Supplementary angles have degree measures that total $180^{\circ}$ , so an angle supplementary to $68^{\circ}$ would measure $180^{\circ}-68^{\circ}=112^{\circ}$ .

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Question

What is the measure of an angle that is complementary to a $42^{\circ }$ angle?

Answer

Complementary angles have degree measures that total $90^{\circ}$ , so an angle complementary to $42^{\circ}$ would measure $90^{\circ}-42^{\circ}=48^{\circ}$ .

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Question

What is the measure of an angle congruent to a $58^{\circ}$ angle?

Answer

Congruent angles have degree measures that are equal, so an angle congruent to $58^{\circ}$ is $58^{\circ}$ .

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Question

What is the measure of an angle that is supplementary to a $80^{\circ}$ angle?

Answer

Supplementary angles have degree measures that total $180^{\circ }$ . Since we have an $80^{\circ}$ angle, the supplementary angle would measure $180^{\circ}-80^{\circ }=100^{\circ}$

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Question

Which of the following angles is complementary to an $18^{\circ}$ angle?

Answer

Complementary angles have degree measures that total $90^{\circ }$ . Since we have an $18^{\circ}$ angle, the supplementary angle would measure $90^{\circ}-18^{\circ }=72^{\circ}$

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Question

Which of the following angles is congruent to a $119^{\circ}$ angle?

Answer

Congruent angles have the same degree measure, so an angle congruent to a $119^{\circ}$ angle would also measure $119^{\circ}$ .

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Question

What is the measurement of an angle that is congruent to an $89^{\circ }$ angle?

Answer

Two angles are congruent if they have the same degree measure. Therefore, an angle congruent to an $89^{\circ }$ angle also measures $89^{\circ }$ .

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Question

What is the measurement of an angle that is supplementary to a $16^{\circ}$ angle?

Answer

Two angles are supplementary if the total of their degree measures is $180^{\circ }$ . Therefore, an angle supplementary to a $16^{\circ}$ angle measures $180^{\circ }-16^{\circ}=164^{\circ}$ .

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Question

What is the measure of an angle that is complementary to a $70^{\circ}$ angle?

Answer

Two angles are complementary if the total of their degree measures is $90^{\circ }$ . Therefore, an angle complementary to a $70^{\circ}$ angle measures $90^{\circ }-70^{\circ}=20^{\circ}$ .

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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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