GMAT DS : Geometry, Coordinate Geometry

Here is a data sufficiency question. It is a question on slopes of lines and tests basic concepts about lines in geometry and coordinate geometry.

Question

Are lines p (with slope m) and q (with slope n) perpendicular to each other?
1. m + 2 = n
2. m + n = 0

Correct Answer: Choice C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.

Explanatory Answer

If two lines are perpendicular, then the product of the slopes of the two lines will be equal to -1.

In this case, if the product m * n = -1, then the two lines will be perpendicular to each other. If the product is not equal to -1, then they are not perpendicular. We need to assess that conclusively.

Statement 1 m + 2 = n
m could be -1 and n could be 1, in which case the product is -1. Alternatively, m could be 4 and n could be 6 in which case the product is not -1.

As we are not able to conclude using the information in statement 1, it is not sufficient. Choices A and D can be eliminated. We are left with choices B, C or E.

Statement 2 m + n = 0.
m could be -1 and n could be 1 or vice versa. In that case, m * n = -1.
m could be any other number and n could be -m. In that case m * n will not be equal to -1. Hence, statement 2 is also not sufficient. We can eliminate choice B. We are left with choices C or E.

Combining the two statements, we know that m = -n from statement 2. Substituting that in statement 1, we get m + 2 = -m or 2m = -2 or m = -1. Hence, n = 1. Hence, the product m * n = -1.

As the information provided in the two statements is sufficient to answer the question, choice C is the correct answer.

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