Data Sufficiency

Hi there,

I was taking the Kaplan practice test and I got this question wrong. Tried doing it again after my test and still got it wrong. Would someone be able to help?

In the rectangular coordinate system, lines m and n cross at the origin. Is line m perpendicular to line n ?

(1) If the slope of m is y and the slope of n is z, then -yz = 1.

(2) m has a slope of -1, and n passes through the point (-x, -x).

Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by itself is not.
Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by itself is not.
Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, even though NEITHER statement BY ITSELF is sufficient.
EITHER statement BY ITSELF is sufficient to answer the question.
Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question, requiring more data pertaining to the problem.

Answer: D

In my opinion, the first statement is incorrect because it considers all possibilities EXCEPT when the slope of m is 0 (horizontal line) and the slope of n is undefined (vertical line). This is the one and only possibility that satisfies that the two lines are perpendicular but does NOT satisfy the "-yz=1" requirement - this is why I ruled out A. Apparently I should not have done this...not so sure of how to approach this question.

Any help would be appreciated.

Thank you!

krithika1993 wrote:Hi there,

I was taking the Kaplan practice test and I got this question wrong. Tried doing it again after my test and still got it wrong. Would someone be able to help?

In my opinion, the first statement is incorrect because it considers all possibilities EXCEPT when the slope of m is 0 (horizontal line) and the slope of n is undefined (vertical line). This is the one and only possibility that satisfies that the two lines are perpendicular but does NOT satisfy the "-yz=1" requirement - this is why I ruled out A. Apparently I should not have done this...not so sure of how to approach this question.

Any help would be appreciated.

Thank you!

Hi krithika1993.

The point of the question is not to figure out whether Statement 1 applies to ALL perpendicular lines.

The point is to determine whether the two lines IN THE QUESTION are perpendicular.

Since if the lines in the question fit Statement 1, they must be perpendicular, Statement 1 is sufficient for determining that the answer to the question is "Yes."

krithika1993 wrote:In my opinion, the first statement is incorrect because it considers all possibilities EXCEPT when the slope of m is 0 (horizontal line) and the slope of n is undefined (vertical line). This is the one and only possibility that satisfies that the two lines are perpendicular but does NOT satisfy the "-yz=1" requirement - this is why I ruled out A. Apparently I should not have done this...not so sure of how to approach this question.

The portion in red violates the rules of DS.
When we evaluate a statement, only cases that satisfy the statement may be considered.
Thus, when we evaluate statement 1, we are not ALLOWED to consider cases in which -yz ≠ 1.
Only cases in which -yz = 1 may be considered.

The slopes of perpendicular lines have a product of -1.

Statement 1, rephrased:
yz = -1.
Since the slopes of lines m and n have a product of -1, m and n must be perpendicular.
SUFFICIENT.

Ahhh I understand now. I was misinterpreting the question and got a bit confused there. Thank you for clarifying!

what about statement 2?can anyone explain ,plz?

sagarock wrote:what about statement 2?can anyone explain ,plz?

Statement 2: m has a slope of -1, and n passes through the point (-x, -x).

To determine whether the two lines are perpendicular, we need either the relationship between their slopes, which is provided by Statement 1, or their actual slopes.

Statement 2 provides the slope of line m. So we need the slope of line n.

We know from the question that line n goes through the origin, point (0,0).

For ANY non zero x, a line that goes through the origin and through point (-x,-x) has a slope of 1.

(Note: The statement actually does not say that x is non zero. So theoretically (-x,-x) could be the origin, but the OA to this question is D. So I guess we have to go with a non zero x to understand how to get to the OA.)

To see why the slope must be 1, you could work mathematically.

Slope = Rise/Run = (y₂ - y�)/(x₂ - x�) = (-x - 0)/(-x - 0) = 1

Alternatively, you could see that a line that passes through the origin and passes through point (-x,-x) must be a line with points (1,1), (2,2), (3,3) and so on and (-1,-1), (-2,-2), (-3,-3) and so on. Such a line has slope 1.

So Statement 2 provides information that can be used to determine the slopes of both lines.

We could use the slopes to determine whether the lines are perpendicular, but we don't have to because this is a data sufficiency question.

Sufficient.

Hii marty sir,For ANY non zero x, a line that goes through the origin and through point (-x,-x) has a slope of 1. Is this a rule ?

sagarock wrote:Hii marty sir,For ANY non zero x, a line that goes through the origin and through point (-x,-x) has a slope of 1. Is this a rule ?

I consider it more a provable math fact than a rule.

I personally had not considered it before I read this question, and it's not something that I would memorize.

It just makes sense, as I demonstrated in my post.