Interesting fact - Line intersection

[welcome]

Please refer attached question.

OA is A.

[x2suresh]

Please refer attached question.

slope of the line QR is -1

(1) slope =-1
clearly parallel lines don't intersect

sufficient

(2) slope =-1

clearly parallel lines don't intersect

there is one possibility that line QR itself segment of line itself..
But still they are not intersecting each other


D

[welcome]

Suresh,
Thanks for your quick responce, even I thought the same. But surprisingly Answers is A. And it is Manhatten test question(so source is also reliable).

As per my knowledge, parallel lines CAN NOT intersect, but what exlaination says is, if parallel lines are overlaping each other they are intersecting eact other.

Also this break my concept that 2 lines can intersect on ONLY one one point otherwise they are parallel.


Please help.

[luckylucky]

Here is how i solved this question

The line passes through (1,3) and (2,2)

Hence the equation of the line is (y-3) = (2-3)/(2-1)(x - 1)

or x + y = 4

1) It says that the equation of the line is x + y = 4

which is the same as above

that means the lines are one over another and they intersect

Hence sufficient

2) Slope = -1

if the slopes are equal then the lines are parallel

However if two lines are parallel they may or may not intersect

Hence not sufficient


Therefore the answer is 'A'

[welcome]

Luckylucky,
Thanks for your reply, but your conclusion "one over another and they intersect " is doesnt seems OK, in regard where 2 overlapping lines are considered parallel and parallel lines never intersect.

Please explain.

[hk]

However if two lines are parallel they may or may not intersect

Hence not sufficient


Therefore the answer is 'A'

I disagree.. The definition of a Parallel lines states that "2 lines are parallel if they have the same slope and they do not Intersect."

So (i) Basically tells you that the slope is -1, so its parallel to QR and does not intersect => Sufficient
(ii) Slope = -1 is basically the same info as above => Sufficient

So the answer is D.

An additional note here regarding the overlapping: when 2 lines overlap they intersect at an infinite number of points. Now even if we agree that overlapping lines intersect at any given point, the answer is going to be the same as both (i) and (ii) will tell you the same exact thing that yes it intersects. So even if you consider this aspect the Answer to the question is D

So can anybody explain why the OA is A clearly?!!

[Stuart@KaplanGMAT]

Let's turn to the Official Guide to help us out.

Page 128 says:

"If two lines that are in the same plane do not intersect, the two lines are parallel."

In this thread, that rule has been interpreted as:

"If two lines that are in the same plane are parallel, the two lines do not intersect."

This interpretation is NOT necessarily true.

Just because "if something is a dog, then it's a mammal", does that mean that "if something is a mammal, it's a dog"? Of course not!

In fact, parallel lines do intersect if they are the same line - they intersect at every point.

So, is it possible for a line segment to intersect with a line that has the same slope? Yes, if the line segment is part of the line itself.

(1) gives us the equation of the line. Well, if we know the equation of the line, we could certainly plot it and see if it intersects the segment: sufficient.

(2) merely gives us the slope of the line. Without knowing where the line lies, we cannot determine if it intersects the line segment: insufficient.

(1) is sufficient alone, (2) is not: choose (A).

[Uri]

Great answer, Stuart. This same logic also applies while solving equations. In 4.2.2 (page 121 of OG 11), it is said that two equivalent equations have infinite number of solutions. And obviously two equivalent equations represent two straight lines where one falls on the other. Good revision of fundamentals!!!

[welcome]

Thanks Stuart and Uri, Great explanation.