GMAT Math : Perpendicular Lines

Study concepts, example questions & explanations for GMAT Math

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Example Questions

Example Question #11 : Perpendicular Lines

Consider the lines of the equations

and 

Are these two lines parallel, perpendicular, or neither?

Statement 1: 

Statement 2: 

Possible Answers:

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.

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 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  , 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  - or, equivalently, , then each of the slopes  and  is the opposite of the reciprocal of the other. This makes the lines perpendicular.

Example Question #12 : Perpendicular Lines

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 .

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 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.

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.

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 .

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 , 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.

Example Question #104 : Coordinate Geometry

Are linear equations  and  perpendicular?

I)  pass through the points  and .

II)  passes through the point  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 

Use II) to find the slope of 

These are not opposite reciprocals, so  and  are not perpendicular.

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