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In the xy-plane, line k passes through the point (1, 1) and line m [#permalink]

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11 Jun 2010, 06:24

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In the xy-plane, line k passes through the point (1, 1) and line m passes through the point (1, -1). Are the lines k and m perpendicular to each other ?

(1) Lines k and m intersect at the point (1, -1) (2) Line k intersects the x-axis at the point (1, 0)

Re: In the xy-plane, line k passes through the point (1, 1) and line m [#permalink]

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11 Jun 2010, 06:33

IMO E.

S1) If line k passes thru (1, 1) and (1, -1) then it is parallel to y axis but we dont know one more different point for line m. Insufficient.

S2) Similar to S1.

S1+S2: Similar to S1, Insufficient.

So, E.
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In the xy-plane ,line k passes through the point (1,1) and line m passes through the point (1,-1).Are the lines k and m perpendicular to each other

(1) Lines k and m intersect at the point (1,-1) (2) Line k intersects the x axis at the point (1,0)

Please help on this DS problem

For one line to be perpendicular to another, their slopes must be negative reciprocals of each other (if slope of one line is \(m\) than the slope of the line perpendicular to this line is \(-\frac{1}{m}\)). In other words, the two lines are perpendicular if and only the product of their slopes is \(-1\).

So basically the question is can we somehow calculate the slopes of these lines.

From stem we have one point for each line.

(1) gives us the second point of line \(k\), hence we can get the slope of this line, but we still know only one point of line \(m\). Not sufficient.

(2) again gives the second point of line \(k\), hence we can get the slope of this line, but we still know only one point of line \(m\). Not sufficient.

(1)+(2) we can derive the slope of line \(k\) but for line \(m\) we still have only one point, hence we can not calculate its slope. Not sufficient.

Answer: E.

For more on this issue please check Coordinate Geometry chapter of Math Book (link in my signature).

Re: In the xy-plane, line k passes through the point (1, 1) and line m [#permalink]

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22 Aug 2010, 17:48

Hey everyone,

Thanks to everyone that responded to my previous round of posts, it was extremely helpful! I have some more questions that I need help with. Here it goes:

In the xy-plane, line k passes through the point (1,1) and line m passes through the point (1,-1). Are lines ka nd m perpendicular to each other?

(1) Lines k and m intersect at the point (1, -1) (2) Line k intersects the x-axis at the point (1,0)

Re: In the xy-plane, line k passes through the point (1, 1) and line m [#permalink]

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22 Aug 2010, 17:53

1. k goes through 1,1 and 1,-1, m can be anyslope. not suff 2. k intersects xaxis at 1,0 <- this probably can be seen from (1) itself. no info about m. not suff 1+2 -> m can be anything. can't say, E
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Re: In the xy-plane, line k passes through the point (1, 1) and line m [#permalink]

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22 Aug 2010, 18:54

uzzy12 wrote:

Thanks for the post but I'm still not grasping the concept. When two lines intersect, doesn't that mean that they are perpendicular to each other?

Yes, when two lines intersect, their angle may or may not be 90 degree. Also we need to find the slope of the two lines (m1 and m2) and for the two lines to be perpendicular, m1 * m2 should be equal to -1.

Since the slope could not be uniquely identified, we cannot conclude that they are perpendicular.
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I did not understand why you said you can find out only line K slope from statement 1 - why not m

sorry but not able to follow please help

Even when considering the statements together we still know only one point of line m: (1, -1). We cannot get the slope based on just one point of a line.

In the xy-plane, line passes through the point (1,1) and line m passes through the point (1,-1). Are lines k and m perpendicular?

1) Lines k and m intersect at the point (1,-1) 2) Line k intersects the x-axis at the point (1,0)

Merging similar topics.

Val1986 wrote:

Could you tell me how to determine whether the lines are perpendicular just in general??

For one line to be perpendicular to another, their slopes must be negative reciprocals of each other (if slope of one line is \(m\) than the slope of the line perpendicular to this line is \(-\frac{1}{m}\)). In other words, the two lines are perpendicular if and only the product of their slopes is \(-1\).

In the xy-plane, line k passes through the point (1, 1) and line m [#permalink]

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14 Jul 2016, 22:12

gautamsubrahmanyam wrote:

In the xy-plane, line k passes through the point (1, 1) and line m passes through the point (1, -1). Are the lines k and m perpendicular to each other ? (1) Lines k and m intersect at the point (1, -1) (2) Line k intersects the x-axis at the point (1, 0)

Two lines are parallel only if their slopes are inverse reciprocal of each other. (1) Lines k and m intersect at the point (1, -1) Using the (x,y) pair of (1,-1) and other (x,y) pair of (1,1) of line k from the question stem we can see that the slope is undefined. Slope of K = \(\frac{1-(-1)}{1-1} = \frac{-2}{0} ==> undefined\) Meaning Line K is parallel to Y axis and pass through the x axis at x=1, y=0 INSUFFICIENT we have no info about the slope of line K

(2) Line k intersects the x-axis at the point (1, 0) We already know from statement 1 that k passes from x=1. INSUFFICIENT

Merging both statement WE NEED SLOPE OF LINE M But there are only one (x,y) pair given for line M and thus we cannot calculate the slope. and thus cannot compare it with slope of line k Merging also doesn't gives any new information Hence insufficient

ANSWER IS E
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In the xy-plane, line k passes through the point (1, 1) and line m passes through the point (1, -1). Are the lines k and m perpendicular to each other ?

(1) Lines k and m intersect at the point (1, -1) (2) Line k intersects the x-axis at the point (1, 0)

Target question: Are lines K and m perpendicular to each other?

IMPORTANT: Since line K passes through the point (1, 1), statements 1 and 2 both have the same effect of "locking" line K into exactly one position. In fact, statements 1 and 2 essentially provide the exact same information. As such, it's either the case that each statement is sufficient (D) or each statement is not sufficient (E).

Since neither statement locks line M into any certain position, line M is free to be in lots of different positions, as long as it passes through the point (1, -1)

Okay, let's jump right to . . . Statements 1 and 2 combined: Here are two possible scenarios that satisfy statements 1 and 2. Scenario a: In this instance, lines M and K are perpendicular.

Scenario b: In this instance, lines M and K are not perpendicular.

Since we cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT