GMAT Math : Geometry

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Example Questions

Example Question #651 : Geometry

Line j passes through the point (4,6) . What is the equation of line j?

1) Line j is perpindicular to the line defined by $y = -\frac{1}{2}x+3$

2) Line j has an x-intercept of

Possible Answers:

Statement 1 alone is sufficient.

Statement 2 alone is sufficient.

Together, the two statements are sufficient.

Neither of the statements, separate or together, is sufficient.

Either of the statements is sufficient.

Correct answer:

Either of the statements is sufficient.

Explanation:

Either statement is sufficient.

Line j, as a line, has an equation of the form y=mx+b

Statement 1 gives the equation of a perpindicular line, so the slopes of the two lines are negative reciprocals of each other:

y=2x+b

6=2(4)+b

b=-2

y=2x-2

Statement 2 allows the slope to be found using rise over run:

$m=\frac{y_2-y_1}{x_2-x_1}=\frac{6-0}{4-1}=3$

Then, since the x-intercept is known:

0=2(1)+b

b=-2

y=2x-2

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

Find the equation for linear function p(x) .

I) $f(x)\parallel p(x)$ and f(x)=16x+5

II) $p(\frac{1}{2})=-4$

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.

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.

Both statements are needed to answer the question.

Correct answer:

Both statements are needed to answer the question.

Explanation:

Find the equation for linear function p(x)

I) $f(x)\parallel p(x)$ and f(x)=16x+5

II) $p(\frac{1}{2})=-4$

To begin:

I) Tells us that p(x) must have a slope of 16

II) Tells us a point on p(x). Plug it in and solve for b:

$-4=\frac{1}{2}*16+b$

b=-12

p(x)=16x-12

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Example Question #764 : Data Sufficiency Questions

Give the equation of a line.

Statement 1: The line interects the graph of the equation $y = x^{2}- 7x + 6$ on the -axis.

Statement 2: The line interects the graph of the equation $y = x^{2}- 7x + 6$ on the -axis.

Possible Answers:

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

EITHER STATEMENT ALONE provides sufficient information to answer the question.

Correct answer:

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

Explanation:

Assume both statements to be true. Then the line shares its - and -intercepts with the graph of $y = x^{2}- 7x + 6$ , which is a parabola. The common -intercept can be found by setting x = 0 and solving for :

$y = x^{2}- 7x + 6$

$y = 0^{2}- 7\cdot 0 + 6 = 0 - 0 + 6 = 6$ ,

making the -intercept of the parabola, and that of the line, (0,6) .

The common -intercept can be found by setting y= 0 and solving for :

$0= x^{2}- 7x + 6$

0 = (x-1)(x-6)

x-1 = 0 , in which case x = 1 , or

x-6 = 0 , in which case x = 6 ,

The parabola therefore has two -intercepts, (1,0) and (6,0) , so it is not clear which one is the -intercept of the line. Therefore, the equation of the line is also unclear.

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

Is the slope of the line Ax + y = C positve, negative, zero, or undefined?

Statement 1: AC > 0

Statement 2: A > 0

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.

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.

Correct answer:

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

Explanation:

Ax + y = C , in slope-intercept form, is

y = -Ax + C

Therefore, the sign of is the sign of the slope.

The first statement means that is positive - all that means is that both and are nonzero and of like sign. can be either positive or negative, and consequently, so can slope .

The second statement - that is positive - makes , the sign of the slope, negative.

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Example Question #766 : Data Sufficiency Questions

Does a given line with intercepts (a,0), (0,b) have positive slope or negative slope?

Statement 1: a = 2b

Statement 2: a = b + 4

Possible Answers:

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT 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 sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient 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 slope of a line through (a,0) and (0,b) is

$m = \frac{b-0}{0-a} =- \frac{b}{a}$

From Statement 1 alone, we can tell that

$m =- \frac{b}{a} =- \frac{b}{2b} = - \frac{1}{2} < 0$ ,

so we know the sign of the slope.

From Statement 2 alone, we can tell that

$m =- \frac{b}{a} =- \frac{b}{b + 4}$

But this can be positive or negative - for example:

$b = 4 \Rightarrow m =- \frac{b}{b + 4}=- \frac{4}{4 + 4} =- \frac{4}{8}= - \frac{1}{2} <0$

but

$b = -2 \Rightarrow m =- \frac{b}{b + 4}=- \frac{-2}{-2 + 4} =- \frac{-2}{2}= 1>0$

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Example Question #43 : Lines

Does a given line with intercepts (a,0), (0,b) have positive slope or negative slope?

Statement 1: 2a + b = 9

Statement 2: a - 2b = 17

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 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.

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:

The slope of a line through (a,0) and (0,b) is

$m = \frac{b-0}{0-a} =- \frac{b}{a}$

If and have the same sign, then m < 0 , making the slope negative; if and have the same sign, then m >0 , making the slope positive.

Statement 1 is not enough to determine the sign of .

2a + b = 9

b = 9 - 2a

Case 1: $a = 1 \Rightarrow b = 9 - 2 \cdot1 = 7$

Case 2: $a = 5 \Rightarrow b = 9 - 2 \cdot5 = -1$

So if we only know Statement 1, we do not know whether and have the same sign, and, subsequently, we do not know the sign of slope . A similar argument can be made that Statement 2 provides insufficient information.

If we know both statements, we can solve the system of equations as follows:

a - 2b = 17

$a - 2 \left ( 9-2a\right ) = 17$

a -18 +4a= 17

5a -18 = 17

5a = 35

a = 7

$b = 9 - 2a = 9 - 2 \cdot 7 = 9 - 14 = -5$

Therefore, we know and have unlike sign and m >0 .

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Example Question #651 : Geometry

Does a given line with intercepts (a,0), (0,b) have positive slope or negative slope?

Statement 1: a = 4b + 5

Statement 2: 3a - 12b = 15

Possible Answers:

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.

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.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

The slope of a line through (a,0) and (0,b) is

$m = \frac{b-0}{0-a} =- \frac{b}{a}$

If and have the same sign, then m < 0 , making the slope negative; if and have the same sign, then m >0 , making the slope positive.

If we know both statements, we try to solve the system of equations as follows:

3a - 12b = 15

3 (4b + 5) - 12b = 15

12b+15 - 12b = 15

15 = 15

This means that the system is dependent, and that the statements are essentially the same.

Case 1: b = 1

Then $a = 4b + 5 = 4 \cdot 1 + 5 = 9$

Case 2: b = -1

Then $a = 4b + 5 = 4 \cdot \left (-1 \right ) + 5 = 1$

Thus from Statement 1 alone, it cannot be determined whether and have the same sign, and the sign of the slope cannot be determined. Since Statement 2 is equivalent to Statement 1, the same holds of this statement, as well as both statements together.

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

You are given two lines. Are they perpendicular?

Statement 1: The sum of their slopes is $\frac{3}{2}$ .

Statement 2: They have the same slope.

Possible Answers:

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.

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:

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

Explanation:

Statement 2 alone tells us that the lines are parallel, not perpendicular. Statement 1 alone is neither necessary nor helpful, as the sum of the slopes is irrelevant.

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

A line includes points (a,b) and (c,d) . Is the slope of the line positive, negative, zero, or undefined?

Statement 1: a-c > b-d

Statement 2: b>d

Possible Answers:

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.

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.

Correct answer:

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

Explanation:

The slope of the line that includes points (a,b) and (c,d) is $m = \frac{b-d}{a -c }$ .

For the question of the sign of the slope to be answered, it must be known whether a-c and b-d are of the same sign or of different signs, or whether one of them is equal to zero.

Statement 1 alone does not answer this question, as it only states that the denominator is greater; it is possible for this to happen whether both are of like sign or unlike sign. Statement 2 only proves that b-d > 0 - that is, that the denominator is positive.

If the two statements together are assumed, we know that a-c > b-d > 0 . Since both the numerator and the denominator are positive, the slope of the line must be positive.

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Example Question #652 : Geometry

Is the slope of a line on the coordinate plane positive, zero, negative, or undefined?

Statement 1: The line is perpendicular to the -axis.

Statement 2: The line 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 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.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

The -axis is horizontal, so any line perpendicular to it is vertical and has undefined slope. Statement 1 is sufficient.

A line on the coordinate plane with no -intercept does not intersect the -axis and therefore must be parallel to it - subsequently, it must be vertical and have undefined slope. This makes Statement 2 sufficient.

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GMAT Math : Geometry

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #651 : Geometry

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

Example Question #764 : Data Sufficiency Questions

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

Example Question #766 : Data Sufficiency Questions

Example Question #43 : Lines

Example Question #651 : Geometry

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

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

Example Question #652 : Geometry

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