GMAT Math : Coordinate Geometry

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Example Questions

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

Calculate the slope of a line perpendicular to line .

Line passes through points and .
The equation for line is .

Possible Answers:

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

Statements 1 and 2 are not sufficient, and additional data is needed 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.

Each statement alone is sufficient to answer the question.

Correct answer:

Each statement alone is sufficient to answer the question.

Explanation:

Statement 1: We can use the points provided to find the slope of line AB.

$\frac{y_{2}-y_{1}}{x_{2}-x_{1}}= \frac{9-5}{1+3}=\frac{4}{4}=1$

Since the slope we're being asked for is of a line perpendicular to line AB, their slopes are inverses of each other.

The slope of our line is then $m=-\frac{1}{m_{AB}}=-\frac{1}{1}=-1$

Statement 2: Since we're provided with the line's equation, we just need to look for the slope.

y=mx+b

Where is the slope and is the y-intercept.

In this case, we have y=x+8 so $m_{AB}=1$ . Because our line is perpendicular to line AB, the slope we're looking for is m=-1

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Example Question #2 : Dsq: Calculating The Slope Of A Perpendicular Line

Find the slope of a line perpendicular to g(t) .

I) g(t) passes through the points (9,6) and (4,-13) .

II) g(t) does not pass through the origin.

Possible Answers:

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.

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

Both statements are needed 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:

Find the slope of a line perpendicular to g(t)

I) g(t) passes through the points (9,6) and (4,-13)

II) g(t) does not pass through the origin

Perpendicular lines have opposite reciprocal slopes. For instance: a line with a slope of $\frac{1}{4}$ would be perpendicular to a line with slope of .

To find the slope of a line, we just need two points.

I) Gives us two points on g(t). We could find the slope of g(t) and then the slope of any line perpendicular to g(t).

$\frac{-13-6}{4-9}=\frac{-19}{-5}=\frac{19}{5}$

So the slope of a line perpendicular to g(t) is equal to:

$\frac{-5}{19}$

II) Is irrelevant or at least not helpful.

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

Consider f(x): 4y=16x-5 and h(x):4y=mx-8 .

Find the slope of h(x) .

I) h(x) passes through the point (0,-2) .

II) h(x) is perpendicular to f(x) .

Possible Answers:

Both statements together are needed 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.

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

Either statement alone is sufficient to answer the question.

Correct answer:

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

Explanation:

We are given a line, f(x), and asked to find the slope of another line, h(x).

I) Gives a point on h(x). We could plug in the point and solve for our slope. When we do this since x=0 we are unable to find the value for our slope. Therefore, statement I is not sufficient to solve the question.

II) Tells us the two lines are perpendicular. Take the opposite reciprocal of the slope of f(x) to find the slope of h(x).

Therefore,

$\\f(x): 4y=16x-5 \\y=4x-\frac{5}{4} \\m=4$

and thus the slope of h(x) will be,

$m_h=-\frac{1}{m_f}=-\frac{1}{4}$ .

Statement II is sufficient to answer the question.

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Example Question #721 : Data Sufficiency Questions

Line AB is perpindicular to Line BC. Find the equation for Line AB.

1. Point B (the intersection of these two lines) is (2,5).

2. Line BC is parallel to the line y=2x.

Possible Answers:

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

Each statement alone is sufficient.

Statements 1 and 2 together are not 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.

Correct answer:

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

Explanation:

To find the equation of any line, we need 2 pieces of information, the slope of the line and any point on the line. From statement 1, we get a point on Line AB. From statement 2, we get the slope of Line BC. Since we know that AB is perpindicular to BC, we can derive the slope of AB from the slope of BC. Therefore to find the equation of the line, we need the information from both statements.

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Example Question #8 : Lines

Given g(x) , find the equation of h(x) , a line $\small \perp$ to g(x) .

$\small g(x)=-13x+5$

I) h(5)=0 .

II) The -intercept of h(x) is at $-\frac{5}{13}$ .

Possible Answers:

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.

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.

Correct answer:

Either statement is sufficient to answer the question.

Explanation:

To find the equation of a perpendicular line you need the slope of the line and a point on the line. We can find the slope by knowing g(x).

I) Gives us a point on h(x).

II) Gives us the y-intercept of h(x).

Either of these will be sufficient to find the rest of our equation.

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Example Question #2 : Perpendicular Lines

Calculate the equation of a line perpendicular to line .

The equation for line is .
Line goes through point .

Possible Answers:

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.

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.

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 #3 : Perpendicular Lines

Find the equation to a line perpendicular to line .

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

Possible Answers:

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

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.

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.

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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GMAT Math : Coordinate Geometry

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

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

Example Question #2 : Dsq: Calculating The Slope Of A Perpendicular Line

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

Example Question #721 : Data Sufficiency Questions

Example Question #8 : Lines

Example Question #2 : Perpendicular Lines

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 #3 : Perpendicular Lines

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