Data Sufficiency

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

(1) Line n passes through the point (-a, -a), where a ≠ 0, and line m has a slope of -1.

(2) The product of the slope of line m and the slope of line n is -1.

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

(1) Line n passes through the point (-a, -a), where a ≠ 0, and line m has a slope of -1.

(2) The product of the slope of line m and the slope of line n is -1.

The rule tested here: If two lines are perpendicular, their slopes will be negative reciprocals.

S1: We know that both lines go through the origin, or point (0,0.) If line n passes through (0,0) and (-a,-a) then its slope (change in y/change in x)
is (-a-0)/(-a-0) = -a/-a = 1. If line n has a slope of 1 and line m has a slope of -1, then the slopes are negative reciprocals, and we can conclude, definitively, that these lines are perpendicular. Statement 1 is sufficient.

S2: If the product of two slopes is -1, those slopes must be negative reciprocals. (It's easy to see this algebraically. Call the slope of Line n: x and the slope of Line m: y. If xy = -1 then x = -1/y, which means that x and y are negative reciprocals.) Statement 2 is also sufficient.

Answer is D.

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

(1) Line n passes through the point (-a, -a), where a ≠ 0, and line m has a slope of -1.

(2) The product of the slope of line m and the slope of line n is -1.

We have 2 variables, the slope and the y-intercept for the line. When two lines are orthogonal, the multiple of their slopes is -1.

This problem has 2 lines, thus 4 variables and thus E is likely the answer. When using both 1) and 2), (1) = (2), making D the answer for almost 95%. Since (1) the line go through the origin, the slope of line n is 1 and the slope of line m is -1, satisfying the orthogonality of the two.


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