In the rectangular coordinate system, lines m and n cross at

Question

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).

mikemcgarry · Accepted Answer

Dear b00gigi , I'm happy to help with this. :-)

You may find these blogs helpful: https://magoosh.com/gmat/2012/gmat-math- ... x-y-plane/ https://magoosh.com/gmat/2012/gmat-math- ... lar-lines/ In this problem, we know the two lines pass through the origin, (0, 0). That's significant. Statement #1 If the slope of m is y and the slope of n is z, then –yz = 1 . Perpendicular lines in the x-y plan have slopes that are negative reciprocals . For example, If one has a slope = 7, the perpendicular line has a slope of –1/7 If one has a slope = 3/5, the perpendicular line has a slope of –5/3 If one has a slope = –4, the perpendicular line has a slope of +1/4 If one has a slope = –8/11, the perpendicular line has a slope of +11/8 Another way to say that is: the product of the slopes of perpendicular lines is -1. Well, if –yz = 1, then yz = –1, and the lines are perpendicular. This statement, alone and by itself, was enough to answer the prompt question, so it is sufficient . Statement #2 m has a slope of –1, and n passes through the point (–x, –x). OK, so the slope of line m is –1. Put that on hold. If n passes through (–x, –x) and (0, 0), then change in y equals change in x, and the slope is +1. Thus, it is perpendicular to line m. This statement, alone and by itself, was enough to answer the prompt question, so it is sufficient . Answer = (D) Does all this make sense? Mike :-)

monirjewel · Answer

Very helpful explanation......thanks for sharing links

jabhatta2 · Answer

Bunuel   karishma  - for S2 - i thought S2 was INS

Slope of m : -1 
Slope of n : calculated as + 1

But how do you know slope of n is "Reciprocal" of slope of m ?

All we know is that the slope of n (positive 1) is "negative" of the slope of m (-1)

Fdambro294 · Answer

Lines M and N will be Perpendicular to each other if the slopes are Negative Reciprocals of each other.

the definition of a Reciprocal of a Number X is the Number that, when multiplied with X, will result in a quotient = 1

the definition of a NEGATIVE Reciprocal of a Number X is the Number that, when multiplied with X, will result in a quotient = (-)1

or ------->(Slope of Line M) * (Slope of Line N) = -1

Are Lines M and N Perpendicular to each other?

(Statement 1) -Y * Z = 1

where Y = Slope of Line m

and Z = Slope of Line n

-Y * Z = 1

Y * Z = -1

YES - the slopes of the 2 Lines are (-)Negative Reciprocal of each other

(statement 2) Line m has a slope of -1, and Line n passes through the Point (-X, -X)

we are also GIVEN that both Lines pass through the Origin (0 , 0)

this means the Y-Intercept for both Lines will be at Y-Coordinate = 0

thus, each Line's Equation will take the Form of: y = (m) * x + 0

if Line m has a slope of -1 -------> then Line m's equation will be: y = (-1) * x

the Line given by y = (-1)*x ----> will be a Downward sloping Line that creates a 45 degree angle with the X-axis in a clockwise direction

points that will fall on Y = (-1) * x will be points such as: (-5 , 5) ; (-4 , 4) ; (0 , 0) ; (1 , -1) (3 , -3)

Each Coordinate will take the Form of: (+X , -Y) -or- (-X , +Y) ----> meaning the coordinates will be Opposites of each other

If Line n passes through the Point (-x , -x) and also passes through the Origin (0 , 0) -----> the slope of this Line will be +1

the equation of Line n will be: y = (+1) * x

the following are examples of Points that will fall on this Line: (-4 , -4) ; (-2 , -2) ; (0 , 0) ; (3 , 3) ; (6 , 6)

this Line too will form a 45 degree angle with the X-axis, but in a COUNTER-clockwise diretion

thus when Line m and Line N intersect through the Origin, they will create a 90 degree angle between the Rays of each Line

or

Slope of Line m = -1

Slope of Line n = +1

and the 2 slopes are (-)negative reciprocals of each other, meaning they will be perpendicular

S2 Sufficient

Answer D

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In the rectangular coordinate system, lines m and n cross at

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