GMAT Math : Geometry

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Example Questions

Example Question #1 : Calculating The Slope Of A Perpendicular Line

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

y=-5x-3

Possible Answers:

$-\frac{1}{25}$

$-\frac{1}{5}$

$\frac{1}{5}$

Correct answer:

$\frac{1}{5}$

Explanation:

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

y=mx+b

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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Example Question #1 : Calculating The Slope Of A Perpendicular Line

What is the slope of a line perpendicular to the line of the equation y = 8 ?

Possible Answers:

$-\frac{1}{ 8}$

$\frac{1}{ 8}$

The line has an undefined slope.

Correct answer:

The line has an undefined slope.

Explanation:

The graph of y = b 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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Example Question #6 : Calculating The Slope Of A Perpendicular Line

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 5x+7y = 0 .

Statement 2: The line is perpendicular to the line of the equation 5x+7y = 0 .

Possible Answers:

EITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Correct answer:

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Explanation:

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 y = 0 and solving for :

5x+7y = 0

5x+7 (0) = 0

5x+0 = 0

5x=0

x = 0

The -intercept of the line is at the origin, (0,0) . 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 5x+7y = 0 can be calculated by putting it in slope-intercept form y = mx+b :

5x+7y = 0

5x+7y -5x = 0 -5x

7y = -5x

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

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

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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Example Question #61 : Coordinate Geometry

Two perpendicular lines intersect at point (5, Y) . One line passes through (3,7) ; the other, through (10, 4) . What is the value of ?

Possible Answers:

$Y = 3 \textrm{ or }Y=6$

$Y = 2 \textrm{ or }Y=14$

$Y = 2 \textrm{ or }Y=9$

$Y = 1 \textrm{ or }Y=18$

$Y = 4\textrm{ or }Y=7$

Correct answer:

$Y = 2 \textrm{ or }Y=9$

Explanation:

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 :

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

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

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

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

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

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

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

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

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

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Example Question #901 : Gmat Quantitative Reasoning

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

Possible Answers:

-0.8, 1.25

-1.75, 1.75

undefined,

0, -1

3.2, 0.3125

Correct answer:

-0.8, 1.25

Explanation:

We can eliminate the choice 3.2, 0.3125 immediately since the slopes of two perpendicular lines cannot have the same sign. We can also eliminate 0, -1 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 -0.8, 1.25 is the correct choice.

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Example Question #661 : Geometry

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

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

Possible Answers:

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

$y=\frac{4}{9}x-13$

$y=\frac{3}{2}x-9$

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

$y=-\frac{4}{9}x+6$

Correct answer:

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

Explanation:

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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Example Question #61 : Lines

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

Possible Answers:

$\frac{1}{2}$

$-\frac{1}{2}$

None of the answers provided

Correct answer:

Explanation:

For a given line defined by the equation y=mx+b 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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Example Question #904 : Problem Solving Questions

What is the slope of any line that is perpendicular to y=7x+3 ?

Possible Answers:

None of the above

$\frac{1}{7}$

$-\frac{1}{7}$

Correct answer:

$-\frac{1}{7}$

Explanation:

For a given line defined by the equation y=mx+b 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=7 , then the slope of any line perpendicular to is $-\frac{1}{m}=-\frac{1}{7}$ .

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Example Question #905 : Problem Solving Questions

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

Possible Answers:

$-\frac{3}{2}$

$\frac{3}{2}$

$-\frac{2}{3}$

$\frac{2}{3}$

None of the above

Correct answer:

$-\frac{3}{2}$

Explanation:

For a given line defined by the equation y=mx+b 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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Example Question #661 : Geometry

Solve for in the coordinate (3,y) on line 5x+2y=13 ?

Possible Answers:

Correct answer:

Explanation:

To solve for for x=3 , we have to plug into the variable of the equation and solve for :

5(3)+2y=13

15+2y=13

2y=13-15

2y=-2

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GMAT Math : Geometry

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #1 : Calculating The Slope Of A Perpendicular Line

Example Question #1 : Calculating The Slope Of A Perpendicular Line

Example Question #6 : Calculating The Slope Of A Perpendicular Line

Example Question #61 : Coordinate Geometry

Example Question #901 : Gmat Quantitative Reasoning

Example Question #661 : Geometry

Example Question #61 : Lines

Example Question #904 : Problem Solving Questions

Example Question #905 : Problem Solving Questions

Example Question #661 : Geometry

All GMAT Math Resources

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