GMAT Math : Coordinate Geometry

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Example Questions

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:

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.

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.

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:

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 #102 : Coordinate Geometry

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:

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.

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

EITHER statement ALONE is 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 #13 : Perpendicular Lines

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:

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.

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

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.

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 : Parallel Lines

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:

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

each statement alone is sufficient

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

both statements taken together are sufficient to answer the question, but neither 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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Example Question #1 : Dsq: Calculating The Equation Of A Parallel Line

Find the equation of the line parallel to the following line:

$\small 5y=-25x+85$

I) The new line passes through the point (-4,12) .

II) The new line has a -intercept of .

Possible Answers:

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

Either statement is sufficient to answer the question.

Explanation:

To find the equation of a parallel line, we need the slope and the y-intercept.

Parallel lines have the same slope, so we have that.

I and II each give us a point on the graph, so we could find the equation of the line through either of them.

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Example Question #743 : Data Sufficiency Questions

Find the equation of the line .

The slope of line is .
Line goes through point .

Possible Answers:

Each statement alone is sufficient to answer the question.

Statement 2 alone is sufficient, but statement 1 alone is not 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.

Statement 1 alone is sufficient, but statement 2 alone is not 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 slope line AB, because we are ask for the equation of the line we need more than just the slope of the line. Therefore, this information alone is not sufficient to write an actual equation.

Statement 2: Using the information from statement 1 and the points provided in this statement, we can answer the question.

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

y-(-6)=(-4)(x-2)

y+6=-4x+8

y=-4x+2

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

Given j(t) , find the equation of h(t) .

j(t)=-12t+789

I) $j(t)\parallel h(t)$

II) h(t) passes through the point (-206,768)

Possible Answers:

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.

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

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:

We are asked to find the equation of a line related to another line.

Statement I tells us the two lines are parallel. This means they have the same slope

Statement II gives us a point on our desired line. We can use this to find the line's y-intercept, which will then allow us to write its equation.

Plug all of the given info into slope-intercept form and solve for b, the line's y-intercept:

$768=-12\cdot-206+b$

b=3240

So our equation is:

h(t)=-12t+3240

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Example Question #21 : Lines

You are given two lines. Are they parallel?

Statement 1: The product of their slopes is .

Statement 2: One has positive slope; one has negative slope.

Possible Answers:

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

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient 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:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

Two parallel lines must have the same slope. Therefore, the product of the slopes will be the product of two real numbers of like sign, which must be positive. Each of the two statements contradicts this, so either statement alone answers the question.

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Example Question #103 : Coordinate Geometry

One line includes the points (A, B) and (1, 3) ; a second line includes the points (-A, -B) and (2, 5) . If these lines are parallel, what is the value of ?

1) B = 11

2) B = 15 - A

Possible Answers:

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

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.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

The lines are parallel, so their slopes are equal.

The slope of the first is $m_{_{1}} = \frac{B - 3}{A - 1}$ .

The slope of the second is $m_{_{2}} = \frac{-B - 5}{-A - 2}$ .

Set the two equal to each other:

$\frac{B - 3}{A - 1}= \frac{-B - 5}{-A - 2}$

If you know that B = 11 , then you can easily find by substituting:

$\small \frac{11 - 3}{A - 1}= \frac{-11 - 5}{-A - 2}$

$\small \small \frac{8}{A - 1}= \frac{-16}{-A - 2}$

Cross-multiply and solve:

$\small 8(-A-2) = -16(A-1)$

$\small -8A-16 = -16A+16$

$\small 8A = 32$

$\small A = 4$

If you know that B = 15 - A , do the same thing:

$\small \frac{15-A - 3}{A - 1}= \frac{-(15-A) - 5}{-A - 2}$

$\small \small \small \frac{-A +12}{A - 1}= \frac{A-20}{-A - 2}$

$\small (-A+12)(-A-2) = (A-1)(A-20)$

$\small \small A^{2}-10A-24= A^{2}-21A+20$

$\small -10A-24= -21A+20$

$\small 11A=44$

$\small A = 4$

Therefore, either statement alone is sufficient to answer the question.

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GMAT Math : Coordinate Geometry

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #101 : Coordinate Geometry

Example Question #102 : Coordinate Geometry

Example Question #13 : Perpendicular Lines

Example Question #104 : Coordinate Geometry

Example Question #1 : Parallel Lines

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

Example Question #743 : Data Sufficiency Questions

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

Example Question #21 : Lines

Example Question #103 : Coordinate Geometry

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