MGMAT CAT 3 :: The incredible Line

[nitin86]

In the xy-coordinate system, what is the slope of the line that goes through the origin and is equidistant from the two points P = (1, 11) and Q = (7, 7)?

a. 2
b. 2.25
c. 2.50
d. 2.75
e. 3

The explaination given is
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The question asks us to find the slope of the line that goes through the origin and is equidistant from the two points P=(1, 11) and Q=(7, 7). It's given that the origin is one point on the requested line, so if we can find another point known to be on the line we can calculate its slope. Incredibly the midpoint of the line segment between P and Q is also on the requested line, so all we have to do is calculate the midpoint between P and Q! (This proof is given below).

Let's call R the midpoint of the line segment between P and Q. R's coordinates will just be the respective average of P's and Q's coordinates. Therefore R's x-coordinate equals 4 , the average of 1 and 7. Its y-coordinate equals 9, the average of 11 and 7. So R=(4, 9).

Finally, the slope from the (0, 0) to (4, 9) equals 9/4, which equals 2.25 in decimal form.

==========================

But, this assumption that "if two points are equidistant from a line, then mid-point will always lie on the line" is INCORRECT. Because, it depends whether the two points that are equidistant from the line are on the same side of the line or different side.

a) If the two points are on the same side of the line, than in that case, both the points lie on a line that is parallel to the given line.

b) If the two points are on different side of the line, then the above reasoning that the mid-point will lie on the line holds true.

But as the question stem does not talk about it, I don't think we can take this assumption.

Tutors, please help.

[Guest]

It is quite safe to assume in this case that the points DO NOT lie on the same side of the line because if so, the slope of the line on which they lie would have been equal to the slope of the line which we need to determine. If you calculate the slope of the line on which these points were to lie, you would see that the slope is -ve (-2/3), whereas you can see from the solutions given that there is no -ve slope, so you can safely assume that the points lie on either side of the line. Please correct me if I am wrong.

[esledge]

First a general comment: In a testing situation I would solve this by drawing a picture (maybe even by drawing the 5 line options from the choices to see which "works"), without getting too theoretical about it.

About your concern: Both posters are right about the two possible lines. On PS, it is OK for there to be two solutions, only one of which is listed. For an algebra example, see OG #171, which uses the words "...which of the following could be the value..." rather than the more definite "...which of the following is the value..."

Perhaps this one could use a wording change from "...what is the slope of the line that..." to "...what is the slope of a line that..." We'll talk it over and make a judgement.

Thanks for your attention to detail!

[malikrulzz]

Can you please explain this again. I haven't got this at-all

[JonathanSchneider]

I'll do my best, but if you're still not getting it, then trust me, this is NOT the most important use of your time : )

Basically, the first poster caught something in the wording of our problem. We asked for the slope of the line that was equidistant from two given points. In other words, the line that crosses through the midpoint between those two points. However, the first poster is correct that this is not the only such line. The perpendicular bisector of that line would ALSO be equidistant from those two points. Thus, the wording on the problem should change to indicate that the given solution is one possible answer, though not the only possible one.

[editor (ron): there is another such line, but it's not the perpendicular bisector of this line. that line wouldn't pass through the origin, and so wouldn't satisfy the criteria of the problem.
the mysterious "other line" would be through the origin and would be parallel to the segment between the two given points, IE slope -2/3. since this line would be parallel to PQ, it would be guaranteed to be equidistant from P and Q.
note that there is no necessary relationship between the slopes of the two lines that solve this problem. they certainly don't have to be perpendicular to each other.
finally, also, it's dicey to talk about the "bisector" of a line. lines don't stop, and so they don't have bisectors. jonathan is actually talking about the bisector of a segment passing through the indicated points, although he didn't mention that explicitly.]

[gagansb]

I'd like to point out something here...
We all know that whenever we are calculating the distance of a point from a line, we take the perpendicular distance.
Now we also know, that there will be a line, parallel to the line throught these points, and passing through 0,0. this line will have a slope of -2/3.

What i dont get is, that if the line passing through the midpoint of these points is indeed the "line passing through 0,0 and equidistant from these points" shouldn't the slope of this line and the slope we found in the previous step follow -> m1m2= -1

I think the approach used is the wrong one, since a line passing through the midpoint of two given points and passing through 0,0 might not necessarily be perpendicular to the line through these points
Please Clarify

[RonPurewal]

I think the approach used is the wrong one, since a line passing through the midpoint of two given points and passing through 0,0 might not necessarily be perpendicular to the line through these points
Please Clarify

to which "approach" are you referring here?

i don't see any approach claiming that the desired line must be perpendicular to segment PQ. indeed, the posted solution is most definitely not perpendicular to that segment.

please clarify. thanks.