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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).
(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).
Posts: 4,475
b00gigi
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
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Mike McGarry
Magoosh Test Prep
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Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)
General Discussion
monirjewel
Joined: 06 Feb 2010
Last visit: 18 Oct 2014
Posts: 122
Concentration: Marketing, Leadership
Schools: University of Dhaka - Class of 2010
GPA: 3.63
WE:Business Development (Consumer Packaged Goods)
Schools: University of Dhaka - Class of 2010
Posts: 122
Very helpful explanation......thanks for sharing links
Posts: 1,324
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
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
b00gigi
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