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Line segment with same slope as a line, do they intersect?

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Line segment with same slope as a line, do they intersect?

by Scott2010 » Thu Dec 02, 2010 1:45 pm
Does line S intersect line segment QR?

Graph shows a line segment from (1,3) to (2,2)

(1) The equation of line S is y = -x + 4.

(2) The slope of line S is -1.

Please help with answer explaination.

[spoiler]Lines are said to intersect if they share one or more points. In the graph, line segment QR connects points (1, 3) and (2, 2). The slope of a line is the change in y divided by the change in x, or rise/run. The slope of line segment QR is (3 - 2)/(1 - 2) = 1/-1 = -1.

(1) SUFFICIENT: The equation of line S is given in y = mx + b format, where m is the slope and b is the y-intercept. The slope of line S is therefore -1, the same as the slope of line segment QR. Line S and line segment QR are parallel, so they will not intersect unless line S passes through both Q and R, and thus the entire segment. To determine whether line S passes through QR, plug the coordinates of Q and R into the equation of line S. If they satisfy the equation, then QR lies on line S.

Point Q is (1, 3):
y = -x + 4 = -1 + 4 = 3
Point Q is on line S.

Point R is (2, 2):
y = -x + 4 = -2 + 4 = 2
Point R is on line S.

Line segment QR lies on line S, so they share many points. Therefore, the answer is "yes," Line S intersects line segment QR.

(2) INSUFFICIENT: Line S has the same slope as line segment QR, so they are parallel. They might intersect; for example, if Line S passes through points Q and R. But they might never intersect; for example, if Line S passes above or below line segment QR.

The correct answer is A.

I did a internet search and found this site helpful, https://www.mathopenref.com/coordintersection.html
In using thier tool, it indicates that two overlapping lines do not intersect. That makes sense to me. I can't figure out why A is sufficient.
[/spoiler]
Last edited by Scott2010 on Fri Dec 03, 2010 9:39 am, edited 1 time in total.
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by Night reader » Thu Dec 02, 2010 1:53 pm
Scott2010 wrote:Does line S intersect line segment QR?

Graph show a line segment from (1,3) to (2,2)

(1) The equation of line S is y = -x + 4.

(2) The slope of line S is -1.
st(1) plug x into equation and test for y. Sufficient
st(2) y=ax+b where a=-1 so y=-x+b; y intercept is unknown. Insufficient

Pick A.
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by Scott2010 » Thu Dec 02, 2010 2:01 pm
Night reader wrote:
Scott2010 wrote:[spoiler]st(1) plug x into equation and test for y. Sufficient
st(2) y=-x+b where a=-1 so y=-x+b; y intercept is unknown. Insufficient

Pick A.[/spoiler]
I understand that. Maybe you could explain intersect in more detail. To me I have always though that intersect means that a line crosses another line. In the example they don't cross, they overlap. The explaination says interset is when a line has one or more points in common. So the later part of that sentence means two lines that overlap. It just didn't seem to make logical sense to me.
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by Night reader » Thu Dec 02, 2010 2:12 pm
Scott2010 wrote:
Night reader wrote:
Scott2010 wrote:[spoiler]st(1) plug x into equation and test for y. Sufficient
st(2) y=-x+b where a=-1 so y=-x+b; y intercept is unknown. Insufficient

Pick A.[/spoiler]
I understand that. Maybe you could explain intersect in more detail. To me I have always though that intersect means that a line crosses another line. In the example they don't cross, they overlap. The explaination says interset is when a line has one or more points in common. So the later part of that sentence means two lines that overlap. It just didn't seem to make logical sense to me.
slope functions as an angle of the line = vertical run over horizontal run e.g. ||||_ _ _ _
y intercept functions as the point where line intersects y-abscess

two lines with the same slope can not have identical intercepts (y and x), x is found when y=0

so basically we have either one line (solid) with one slope or parallel lines; intercepts act as intervals on abscess x- and y-

st(2) slope is -1 = line S is open to possibilities of parallel line or the solid (the same line equation as in st. 1). We don't know exactly therefore the terms in st(2) are insufficient.

also, watch the 'line segment' (finite) is different from a line (infinite). Parallel line(s) may go upper or downward area from the line segment - no cross over...
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by goyalsau » Fri Dec 03, 2010 1:41 am
Scott2010 wrote: I did a internet search and found this site helpful, https://www.mathopenref.com/coordintersection.html
In using thier tool, it indicates that two overlapping lines do not intersect. That makes sense to me. I can't figure out why A is sufficient.
Great Post, I found your link very Helpful, Even I thought the same way that when two lines are parallel and if they are collinear they will not intersect but i was wrong, Here is what i found.

With two line segments, we can derive an equation for each, say y=m1x+b1 and y=m2x+b2. The x and y values at the intersection point must satisfy both, so y = m1x+b1=m2x+b2. If we take the last half of this and solve for x, we get m1x+b1=m2x+b2, or (m1-m2)x=b2-b1 or x=(b2-b1)/(m1-m2). Notice that when m1 equals m2, the lines are parallel and they will never intersect. (unless b1 also equals b2 in which case the lines are co-linear).
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by Ian Stewart » Sat May 14, 2011 2:47 am
I received a PM asking for comment on this question. A few points:
Scott2010 wrote: I did a internet search and found this site helpful, https://www.mathopenref.com/coordintersection.html
In using thier tool, it indicates that two overlapping lines do not intersect. That makes sense to me. I can't figure out why A is sufficient.
Nowhere at that link above do they discuss the case of two overlapping lines. Technically two lines intersect if they meet at one or more points, so if two lines completely overlap, they certainly intersect.

That said, the real GMAT would never even allow for the possibility of two completely overlapping lines in a question asking about intersection points. It's the kind of technicality that the GMAT goes to lengths to avoid testing. You won't need to worry about this particular special case on your test, and the question in the original post is not a realistic practice question; you can safely ignore it.
Scott2010 wrote:Does line S intersect line segment QR?

Graph shows a line segment from (1,3) to (2,2)

(1) The equation of line S is y = -x + 4.

(2) The slope of line S is -1.
The solution posted in the original post above is also needlessly long. I'm curious where the question and solution are from. From the stem and Statement 1, we know absolutely everything about both our lines, so we can draw them and answer any question at all about them; there is no information missing, so of course Statement 1 has to be sufficient. You don't need to do any work.

Statement 2 is not sufficient because the two lines might overlap, or they might be parallel non-overlapping lines, but this is not the kind of 'trick' you'll ever see on the GMAT.
For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com

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by sourabh33 » Sat May 14, 2011 4:46 am
Thank you so much for your valuable comments.

Best Regards
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by [email protected] » Fri Mar 16, 2012 10:54 pm
Yes actually, this post was really very important... People who have co-ordinate geometry as a weakness in their prep, they should solve such sums to get an indepth knowledge regarding it...
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