Calculating whether lines are perpendicular

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GMAT Quantitative › Calculating whether lines are perpendicular

Questions 1 - 10
1

Transversal

Figure NOT drawn to scale.

Refer to the above figure.

True or false:

Statement 1: is a right angle.

Statement 2: and are supplementary.

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.

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.

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

Explanation

Statement 1 alone establishes by definition that , but does not establish any relationship between and .

By Statement 2 alone, since same-side interior angles are supplementary, , but no conclusion can be drawn about the relationship of , since the actual measures of the angles are not given.

Assume both statements are true. If two lines are parallel, then any line in their plane perpendicular to one must be perpendicular to the other. and , so it can be established that .

2

Transversal

Refer to the above figure. . True or false:

Statement 1:

Statement 2: and are supplementary.

EITHER statement ALONE is 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 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.

Explanation

If transversal crosses two parallel lines and , then same-side interior angles are supplementary, so and are supplementary angles. Also, corresponding angles are congruent, so .

By Statement 1 alone, angles and are congruent as well as supplementary; by Statement 2 alone, and are also supplementary as well as congruent. Two angles that are both supplementary and congruent are both right angles, so from either statement alone, and intersect at right angles, so, consequently, .

3

Which of the following choices give the slopes of two perpendicular lines?

undefined,

Explanation

We can eliminate the choice immediately since the slopes of two perpendicular lines cannot have the same sign. We can also eliminate and undefined, , since a line with slope 0 and a line with undefined slope are perpendicular to each other, not a line of slope -1 or 1.

Of the two remaining choices, we check for the choice that includes two numbers whose product is -1.

and

so is the correct choice.

4

Find the slope of a line that is perpendicular to the line running through the points and .

Not enough information provided.

Explanation

To find the slope of the line running through and , we use the following equation:

The slope of any line perpendicular to the given line would have a slope that is the negative reciprocal of , or . Therefore,

5

What is the slope of any line that is perpendicular to ?

None of the answers provided

Explanation

For a given line defined by the equation with slope , any line perpendicular to has a slope of , or the negative reciprocal of . Since the slope of the provided line in this instance is , then the slope of any line perpendicular to is .

6

What is the slope of any line that is perpendicular to ?

None of the above

Explanation

For a given line defined by the equation with slope , any line perpendicular to has a slope of , or the negative reciprocal of . Since the slope of the provided line in this instance is , then the slope of any line perpendicular to is .

7

What is the slope of any line that is perpendicular to ?

None of the above

Explanation

For a given line defined by the equation with slope , any line perpendicular to has a slope of , or the negative reciprocal of . Since the slope of the provided line in this instance is , then the slope of any line perpendicular to is .

8

Which of the following lines is perpendicular to

Two of the answers are correct.

Explanation

Given a line defined by the equation with a slope of , any line perpendicular to would have a slope that is the negative reciprocal of , . Given our equation , we know that and that .

There are two answer choices with this slope, and .

9

A given line is defined by the equation . Which of the following lines would be perpendicular to line ?

Not enough information provided

Explanation

For any line with an equation and slope , a line that is perpendicular to must have a slope of , or the negative reciprocal of . Given , we know that and therefore know that .

Only one equation above has a slope of : .

10

Determine whether the lines with equations and are perpendicular.

They are not perpendicular

They are perpendicular

There is not enough information to determine the answer

Explanation

If two equations are perpendicular, then they will have inverse negative slopes of each other. So if we compare the slopes of the two equations, then we can find the answer. For the first equation we have

so the slope is .

So for the equations to be perpendicular, the other equation needs to have a slope of 3. For the second equation, we have

so the slope is .

Since the slope of the second equation is not equal to 3, then the lines are not perpendicular.

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