GMAT Math : Lines

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Example Question #3 : Dsq: Calculating The Equation Of A Perpendicular Line

Calculate the equation of a line perpendicular to line .

The equation for line is .
Line goes through point .

Possible Answers:

Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.

Statements 1 and 2 are not sufficient, and additional data is needed to answer the question.

Statement 1 alone is sufficient, but statement 2 alone is not sufficient to answer the question.

Statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question.

Each statement alone is sufficient to answer the question.

Correct answer:

Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.

Explanation:

Statement 1: We're given the equation to line AB which contains the slope. Because the line we're being asked for is perpendicular to it, we know the slope will be its inverse.

The slope of our line is then $-\frac{1}{2}$

Statement 2: We can write the equation to the perpendicular line only if we have a point that falls within that line. Luckily, we're given such a point in statement 2.

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

$y+7=-\frac{1}{2}(x-4)$

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

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

Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.

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Example Question #4 : Dsq: Calculating The Equation Of A Perpendicular Line

Find the equation of the line perpendicular to r(x) .

I) r(x) has a slope of .

II) The line must pass through the point (9, 96) .

Possible Answers:

Neither statement is sufficient to answer the question. More information is needed.

Statement I is sufficient to answer the question, but statement II is not sufficient to answer the question.

Both statements are needed to answer the question.

Either statement is sufficient to answer the question.

Statement II is sufficient to answer the question, but statement I is not sufficient to answer the question.

Correct answer:

Both statements are needed to answer the question.

Explanation:

Find the equation of the line perpendicular to r(x)

I) r(x) has a slope of -15

II) The line must pass through the point (9, 96)

Recall that perpendicular lines have opposite reciprocal slopes.

Use I) to find the slope of our new line

$m_2=\frac{1}{15}$

Use II) along with our slope to find the y-intercept of our new line.

$96=\frac{1}{15}\cdot 9+b$

b=95.4

$y=\frac{x}{15}+95.4$

Therefore both statements are needed.

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Example Question #5 : Dsq: Calculating The Equation Of A Perpendicular Line

Consider f(t) :

f(t)=13t-8

Find j(t) , a line perpendicular to f(t) , given the following:

I) j(t) passes through the point (14,16) .

II) f(t) passes through the point (2,18) .

Possible Answers:

Either statement is sufficient to answer the question.

Both statements are needed to answer the question.

Neither statement is sufficient to answer the question. More information is needed.

Statement I is sufficient to answer the question, but Statement II is not sufficient to answer the question.

Statement II is sufficient to answer the question, but Statement I is not sufficient to answer the question.

Correct answer:

Statement I is sufficient to answer the question, but Statement II is not sufficient to answer the question.

Explanation:

Recall that perpendicular lines have opposite, reciprocal slopes. We can find the slope of j(t) from the question.

Statement I gives us a point on j(t) , which we can use to find the y-intercept of j(t) , and then the equation.

The slope of j(t) must be the opposite reciprocal of f(t) , this makes our slope $-\frac{1}{13}$ .

Statement I tells us that j(t) passes through the point (14,16) , so we can use slope-intercept form to find our equation:

$16=-\frac{1}{13}(14)+b$

$16=-\frac{14}{13}+b$

$\frac{208}{13}=-\frac{14}{13}+b$

$\frac{222}{13}=b$

$b=17\frac{1}{13}$

So, our equation is

$y=-\frac{1}{13}x+17\frac{1}{13}$

Statement II gives us a point on f(t) , which does not help us in the slightest with j(t) . Therefore, only Statement I is sufficient.

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Example Question #6 : Dsq: Calculating The Equation Of A Perpendicular Line

Give the equation of a line on the coordinate plane.

Statement 1: The line shares an -intercept and its -intercept with the line of the equation 6x+11y = 0 .

Statement 2: The line is perpendicular to the line of the equation 6x+11y = 0 .

Possible Answers:

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

EITHER statement ALONE is sufficient to answer the question.

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

Correct answer:

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

Explanation:

Assume Statement 1 alone. The -intercept of the line of the equation can be found by substituting y = 0 and solving for :

6x+11y = 0

6x+11 (0) = 0

6x+0 = 0

6x=0

x =0

The -intercept of the line is the origin (0,0) ; it follows that this is also the -intercept.

Therefore, Statement 1 alone yields only one point of the line, from which its equation cannot be determined.

Assume Statement 2 alone. The slope of the line of the equation 6x+11y = 0 can be calculated by putting it in slope-intercept form y = mx+b :

6x+11y = 0

11y = -6x

$11y \div 11 = -6x \div 11$

$y = -\frac{6}{11} x$

The slope of this line is the coefficient of , which is $-\frac{6}{11}$ . A line perpendicular to this one has as its slope the opposite of the reciprocal of $-\frac{6}{11}$ , which is

$- \frac{1}{-\frac{6}{11}} = \frac{11}{6}$ .

However, there are infintely many lines with this slope, so no further information can be determined.

Now assume both statements to be true. From Statement 1, the slope of the line is $m = \frac{11}{6}$ , and from Statement 2, the -coordinate of the -initercept is b= 0 . Substitute in the slope-intercept form:

y = mx+b

$y = \frac{11}{6}x+0$

$y = \frac{11}{6}x$

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Example Question #7 : Dsq: Calculating The Equation Of A Perpendicular Line

Find the equation to a line perpendicular to line .

The slope of line is $-\frac{5}{3}$ .
Line goes through point .

Possible Answers:

Statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question.

Statement 1 alone is sufficient, but statement 2 alone is not sufficient to answer the question.

Each statement alone is sufficient to answer the question.

Statements 1 and 2 are not sufficient, and additional data is needed to answer the question.

Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.

Correct answer:

Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.

Explanation:

Statement 1: Since the line we're looking for is perpendicular to line XY, our slope will be the inverse of line XY's slope $-\frac{5}{3}$ .

The slope of our line is then $m=\frac{3}{5}$ . Just knowing the slope however, is not sufficient information to answer the question.

Statement 2: We're provided with a point which will allow us the write the equation.

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

$y-3=\frac{3}{5}(x+5)$

$y-3=\frac{3}{5}x+3$

$y=\frac{3}{5}x+6$

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Example Question #101 : Coordinate Geometry

Consider the lines of the equations

y = A x + 7

and

y = Bx -7

Are these two lines parallel, perpendicular, or neither?

Statement 1: $\frac{A}{B} < 0$

Statement 2: A B = -1

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.

BOTH statements TOGETHER are insufficient to answer the question.

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

Correct answer:

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

Explanation:

Since the two equations are in slope-intercept form, coefficients and are the slopes of the two lines.

If $\frac{A}{B} < 0$ , then this tells us that one of slopes and is positive and one is negative; this only eliminates the possibility of the lines being parallel.

If A B = -1 - or, equivalently, $B = - \frac{1}{A}$ , then each of the slopes and is the opposite of the reciprocal of the other. This makes the lines perpendicular.

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Example Question #2851 : Gmat Quantitative Reasoning

You are given two lines. Are they perpendicular?

Statement 1: The product of their slopes is 1.

Statement 2: The sum of their slopes is $\frac{5}{2}$ .

Possible Answers:

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

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

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

Correct answer:

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

Explanation:

The product of the slopes of two perpendicular lines is , so from Statement 1 alone, from the fact that this product is 1, you can deduce that the slopes are not perpendicular.

Statement 2 is neither necessary nor helpful, since the sum of the slopes is irrelevant to the question.

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Example Question #103 : Coordinate Geometry

Are Line 1 and Line 2 on the coordinate plane perpendicular?

Statement 1: Line 1 is the line of the equation x = 12 .

Statement 2: Line 2 has no -intercept.

Possible Answers:

EITHER statement ALONE is 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 insufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 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.

Correct answer:

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

Explanation:

We need to know the slopes of both lines to answer this question. Each statement gives information about only one line, so neither alone gives a definite answer.

Statement 1 tells us that Line 1 is vertical, since it is a line of the equation x = a , for some real . Statement 2 tells us that the line, not crossing the -axis, must be parallel to the -axis and, subsequently, horizontal. A vertical line and a horizontal line are perpendicular, so the two statements together answer the question.

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Example Question #104 : Coordinate Geometry

Are linear equations g(t) and f(x) perpendicular?

I) g(t) pass through the points (6,7) and (45, 95) .

II) f(x) passes through the point (8,14) and has a -intercept of .

Possible Answers:

Statement II is sufficient to answer the question, but statement I is not sufficient to answer the question.

Neither statement is sufficient to answer the question. More information is needed.

Both statements are needed to answer the question.

Either statement is sufficient to answer the question.

Statement I is sufficient to answer the question, but statement II is not sufficient to answer the question.

Correct answer:

Both statements are needed to answer the question.

Explanation:

To find whether these two functions are perpendicular we need to find each of their slopes.

Perpendicular lines have opposite, reciprocal slopes.

Use I) to find the slope of g(t)

$\frac{95-7}{45-6}=\frac{88}{39}$

Use II) to find the slope of f(x)

$\frac{17-14}{0-8}=\frac{-3}{8}$

These are not opposite reciprocals, so g(t) and f(x) are not perpendicular.

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Example Question #1 : Dsq: Calculating The Equation Of A Parallel Line

Data Sufficiency Question

What is the slope of a line that passes through the point (2,3)?

1. It passes through the origin

2. It does not intersect with the line $y=\frac{3}{2}x+6$

Possible Answers:

statement 1 alone is sufficient, but statement 2 alone is not sufficient to answer the question

statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question

both statements taken together are sufficient to answer the question, but neither statement alone is sufficient

statements 1 and 2 together are not sufficient, and additional data is needed to answer the question

each statement alone is sufficient

Correct answer:

each statement alone is sufficient

Explanation:

In order to calculate the equation of a line that passes through a point, we need one of two pieces of information. If we know another point, we can calculate the slope and solve for the -intercept, giving us the equation of the line. Alternatively, if we know the slope (which we can conclude from the parallel line in statement 2) we can calculate the -intercept and determine the equation of the line.

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GMAT Math : Lines

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #3 : Dsq: Calculating The Equation Of A Perpendicular Line

Example Question #4 : Dsq: Calculating The Equation Of A Perpendicular Line

Example Question #5 : Dsq: Calculating The Equation Of A Perpendicular Line

Example Question #6 : Dsq: Calculating The Equation Of A Perpendicular Line

Example Question #7 : Dsq: Calculating The Equation Of A Perpendicular Line

Example Question #101 : Coordinate Geometry

Example Question #2851 : Gmat Quantitative Reasoning

Example Question #103 : Coordinate Geometry

Example Question #104 : Coordinate Geometry

Example Question #1 : Dsq: Calculating The Equation Of A Parallel Line

All GMAT Math Resources

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