GMAT Math : Geometry

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Example Questions

Example Question #61 : Lines

Veronica's teacher gave her two equations, the first with the coefficients of both variables missing, as follows:

$\square x- \bigcirc y = 9$

5 x - 3 y = 9

Veronica was challenged to write one number in each shape in order to form an equation whose line has the same slope as that of the second equation. The only restriction was that she could not write a 5 in the square or a 3 in the circle.

Did Veronica write an equation with the correct slope?

Statement 1: Veronica wrote a negative integer in the square and a positive integer in the circle.

Statement 2: Veronica wrote an 8 in the circle.

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.

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.

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 the line of

5 x - 3 y = 9

can be found by writing this equation in slope-intercept form:

- 3 y =-5x+ 9

$\frac{- 3 y}{-3} =\frac{-5x+ 9}{-3}$

$y=\frac{-5x }{-3} + \frac{ 9}{-3}$

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

The slope of the line is the coefficient of is $\frac{5}{3}$ , so Veronica must place the numbers in the shapes to yield an equation whose slope has this equation.

Rewrite the top equation as

A x-By = 9

The slope, in terms of and , can be found similarly:

-By =- A x+ 9

$\frac{-By }{-B}=\frac{- A x+ 9 }{-B}$

$y=\frac{- A x }{-B} + \frac{9}{-B}$

$y=\frac{ A }{ B} x- \frac{9}{B}$

Its slope is $\frac{A}{B}$ .

Statement 1 asserts that and are of unlike sign, so the slope $\frac{A}{B}$ must be negative. It cannot have sign $\frac{5}{3}$ , so the question is answered.

Assume Statement 2 alone. Then in the above equation, B=8 , so the slope is $\frac{A}{8}$ . The slope now depends on the value of , so Statement 2 gives insufficient information.

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Example Question #783 : Data Sufficiency Questions

Give the slope of a line on the coordinate plane.

Statement 1: The line passes through the graph of the equation y = |x+1| on the -axis.

Statement 2: The line passes through the graph of the equation y = |x-1| on the -axis.

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.

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 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 -intercept(s) of the graph of y = |x+1| can be found by setting y = 0 and solving for :

y = |x+1|

0= |x+1|

x+1 = 0

x = -1

The graph has exactly one -intercept, (-1,0) .

The -intercept(s) of the graph of y = |x-1| can be found by setting x= 0 and solving for :

y = |x-1|

y = |0-1|

y = | -1|

y = 1

The graph has exactly one -intercept, (0, 1) .

In order to determine the slope of a line on the coordinate plane, the coordinates of two of its points are needed. Each of the two statements yields one of the points, so neither statement alone is sufficient to determine the slope. The two statements together, however, yield two points, and are therefore enough to determine the slope.

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Example Question #20 : Dsq: Calculating The Slope Of A Line

Give the slope of a line on the coordinate plane.

Statement 1: The line passes through the vertex of the parabola of the equation $y = x^{2} + 4$ .

Statement 2: The line passes through the -intercept of the parabola of the equation $y = x^{2} + 4$ .

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:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

The vertex of the parabola of the equation $y = ax^{2}+bx + c$ can be found by first taking $x = -\frac{b}{2a}$ , then substituting in the equation and solving for .

$y = x^{2} + 4$

$x = -\frac{0}{2 \cdot 1} = 0$

$y = x^{2} + 4 = 0^{2} + 4 = 0+4 = 4$

The vertex is the point (0, 4) . Since x = 0 , this is also the -intercept.

In order to determine the slope of a line on the coordinate plane, the coordinates of two of its points are needed. From the two statements together, we only know the -intercept and the vertex; however, they are one and the same. Therefore, we have insufficient information to find the slope.

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Example Question #781 : Data Sufficiency Questions

Consider segment with midpoint at the point (6,7) .

I) Point has coordinates of (4,5) .

II) Segment has a length of units.

What are the coordinates of point ?

Possible Answers:

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.

Either statement alone is sufficient to answer the question.

Both statements are necessary to answer the question.

Neither I nor II are sufficient to answer the question. More information is needed.

Correct answer:

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

Explanation:

In this case, we are given the midpoint of a line and asked to find one endpoint.

Statement I gives us the other endpoint. We can use this with midpoint formula (see below) to find our other point.

Midpoint formula: $\small (x',y')=\left(\frac{x^1-x^2}{2}, \frac{y^1-y^2}{2}\right)$

Statment II gives us the length of the line. However, we know nothing about its orientation or slope. Without some clue as to the steepness of the line, we cannot find the coordinates of its endpoints. You might think we can pull of something with distance formula, but there are going to be two unknowns and one equation, so we are out of luck.

So,

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

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Example Question #782 : Data Sufficiency Questions

Find endpoint given the following:

I) Segment has its midpoint at (45,65) .

II) Point is located on the -axis, points from the origin.

Possible Answers:

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.

Both statements are needed to answer the question.

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

Correct answer:

Both statements are needed to answer the question.

Explanation:

Find endpoint Y given the following:

I) Segment RY has its midpoint at (45,65)

II) Point R is located on the x-axis, 13 points from the origin.

I) Gives us the location of the midpoint of our segment

(45,65)

II) Gives us the location of one endpoint

(13,0)

Use I) and II) to work backwards with midpoint formula to find the other endpoint.

$(45,65)=\left(\frac{x_1-13}{2},\frac{y_1-0}{2}\right)$

$45\cdot2=x_1-13$

x_1=103

$65\cdot2=y_1=130$

So endpoint is at (103,130) .

Therefore, both statements are needed to answer the question.

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Example Question #791 : Data Sufficiency Questions

Consider segment

I) Endpoint is located at the origin

II) has a distance of 36 units

Where is endpoint located?

Possible Answers:

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 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:

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

Explanation:

To find the endpoint of a segment, we can generally use the midpoint formula; however, in this case we do not have enough information.

I) Gives us one endpoint

II) Gives us the length of DF

The problem is that we don't know the orientation of DF. It could go in infinitely many directions, so we can't find the location of without more information.

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Example Question #791 : Data Sufficiency Questions

(2,3) is the midpoint of line PQ. What are the coordinates of point P?

(1) Point Q is the origin.

(2) Line PQ is 8 units long.

Possible Answers:

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Correct answer:

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Explanation:

The midpoint formula is

$\left(\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2}\right)$ ,

with statement 1, we know that Q is $\left ( 0,0\right )$ and can solve for P:

$\frac{0+x}{}2=2$ and $\frac{0+y}{}2=3$

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

x=4 y=6

$p=\left ( 4,6\right )$

Statement 1 alone is sufficient.

Statement 2 doesn't provide enough information to solve for point P.

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Example Question #1 : Dsq: Calculating The Midpoint Of A Line Segment

A line segment has one of its endpoints at (12,13) . In which quadrant, or on what axis, is its other endpoint?

Statement 1: The midpoint of the segment is (9,9) .

Statement 2: The length of the segment is 10.

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:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

Statement 1 give us the means to find the other endpoint using the midpoint formula:

$x _{m } =\frac{x _{1 } + x _{2 }}{2}$

$9 =\frac{13 + x _{2 }}{2}$

$18 =12 + x _{2 }$

$x _{2 } = 6$

Similarly,

$y _{m } =\frac{y _{1 } + y _{2 }}{2}$

$9 =\frac{12 + y _{2 }}{2}$

$18 =13 + y _{2 }$

$y_{2 } = 5$

This makes the endpoint (6,5) , which is in Quadrant I.

Statement 2 is also sufficient. (12,13) , which is in Quadrant 1, is 12 units away from the nearest axis; since the length of the segment is 10, the entire segment must be in Quadrant I.

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Example Question #792 : Data Sufficiency Questions

In what quadrant or axis is the midpoint of the line segment with endpoints (A,B) and (B,A) located?

Statement 1: A > -B

Statement 2: (A,B) is in Quadrant IV.

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 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.

BOTH statements TOGETHER are insufficient 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 midpoint of the segment with endpoints (A,B) and (B,A) is $\left ( \frac{A+B}{2}, \frac{A+B}{2} \right )$ .

If A > -B , then A + B > 0 and $\frac{A+B}{2}>0$ , so the midpoint, having both of its coordinates positive, is in Quadrant I.

If (A,B) is in Quadrant IV, then A >0 and B <0 . But the quadrant of the midpoint varies according to and :

Example 1: If A = 8,B = -4 , the midpoint is $\left ( \frac{8+(-4) }{2}, \frac{8+(-4) }{2} \right )$ , or (2,2) , putting it in Quadrant I.

Example 2: If A = 4,B = -8 , the midpoint is $\left ( \frac{4-8 }{2}, \frac{4-8 }{2} \right )$ , or (-2,-2) , putting it in Quadrant III.

Therefore, the first statement, but not the second, tells us all we need to know.

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Example Question #3 : Dsq: Calculating The Midpoint Of A Line Segment

Consider segment . What are the coordinates of the midpoint of ?

I) Point has coordinates of (3,6) .

II) Point has coordinates of (7,14) .

Possible Answers:

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

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

Each statement alone is enough to solve the question.

Both statements taken together are sufficient to solve the question.

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

Correct answer:

Both statements taken together are sufficient to solve the question.

Explanation:

We are asked to find the midpoint of a line segment and given endpoints in our clues.

Midpoint formula is found by taking the average of the x and y values of two points.

$midpoint=\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2} \right )$

We need both endpoints to solve this problem, so both statements are needed.

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GMAT Math : Geometry

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #61 : Lines

Example Question #783 : Data Sufficiency Questions

Example Question #20 : Dsq: Calculating The Slope Of A Line

Example Question #781 : Data Sufficiency Questions

Example Question #782 : Data Sufficiency Questions

Example Question #791 : Data Sufficiency Questions

Example Question #791 : Data Sufficiency Questions

Example Question #1 : Dsq: Calculating The Midpoint Of A Line Segment

Example Question #792 : Data Sufficiency Questions

Example Question #3 : Dsq: Calculating The Midpoint Of A Line Segment

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