Who is correct?
Between Jim & Jack, who is correct?
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Jim says : '' A point cannot divide a line segment externally in the ratio 1 : 1 ''
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Jack says : " A point cannot divide a line segment externally in the ratio − 1 : 1 "
AYWC?
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1 solution
Discussions for this problem are now closed
Perhaps it would help if you clarified what "dividing a line segment externally" meant.
do you mean in the solution or in the problem??
In the problem. However, if it is too ungainly to write out a long explanation, then I suppose you can leave it be.
@Daniel Liu – Well nothing that... but I preferred not to. Because it can be looked up anywhere. I also posted it under the wiki 'section formula' . So that after people reading the wiki would know what an external division means. One more reason was that, if I explained what external division is I would have to disclose a part of the solution.
Well, everything went better than expected. As I framed the question myself, I worried that no one would attempt it. It may appear uninteresting and too trivial. But unexpectedly I got "362 attempts 110 solvers ". Which ain't bad. This easy looking problems surfaces a concept which if not cleared may cause trouble later.
@Soumo Mukherjee – Yea, I guess it is better not to explain it. I should have known to search up what the term meant.
Good job on those attempt numbers ⌣ ¨
@Daniel Liu – Thanks... I am trying to improve a bit more though. How did you make that smiling face at then end, the only way I know to do it is as :) ... rotating it by right angles :D .
@Soumo Mukherjee – It's L A T E X . You can hover your mouse over it, and it will show you the code:
\ddot\smile
@Soumo Mukherjee – I looked up that the ratio is positive in case of external division. In case definition matters, it's better to clarify what you mean.
@Jakub Šafin – I couldn't follow what you meant to tell?
@Soumo Mukherjee – You said it can be looked up anywhere. I said that I found an explanation that defined the ratio (answer) as positive for external division, not negative - so clarifying what external division and the ratio associated with could be a good idea.
@Jakub Šafin – Okay... well, my problem isn't about whether the ratio is positive or negative, It is about whether the given conditions can be satisfied or not.
If you want to view external division as internal division then the ratio gets its sign changed. If you know the formula to write the coordinates of point which divides a line externally then it's good. And if you apply it you will arrive at the same result. But the formula for the case of internal division is better remembered so solving it by changing into internal division isn't a bad idea. Anyways, by looking up a term's definition if you understand the problem then you really have got the definition used in the context. But if after looking at the definition of a term you still have a doubt and couldn't understand the problem then it is better to get a bit deeper till you figure out what it actually means. In short why would anyone attempt a problem which he doesn't understand?
@Soumo Mukherjee – It's not like I could have a doubt about "is the sign positive or negative?". It either is, within a given theory, or isn't, there's nothing to doubt out of what's written in the problem statement. Understanding doesn't simply divide into right or wrong, there are various kinds of both.
@Jakub Šafin – We both are now drifting away from the topic. Okay I just wanted to say is, we must clarify our doubts before solving a problem. It is not good to proceed having a touchy feeling, even about a definition, solving it. I also referred to a dictionary and found that the dentition of external division, regard to this problem, is insufficient. Because it didn't consider analytic or coordinate geometry. Which makes quite sense because coordinate geometry is not a different branch but a method. However if I explain external division in terms of coordinate, I have to use the section formula for external division. Then the problem will reduce to "given the ratio of division find the coordinates through the explained formula". Would it be a good problem? A boring problem?
My solution isn't the only solution. Some body can do that with pure reasoning without any formula and all. However, my solution simply tell that assume such points exist and proceeds to show that one of them leads to contradiction. I won't drain more of your time. But if you are still not satisfied, plz try to view the problem and the solution together. Then the problem separately. May be, this could be the way where I can convey what I really wanted to. ⌣ ¨
Dividing externally in the ratio ( m : n ) means dividing internally in the ratio ( m : − n )
If P ≡ ( h , k ) divides externally the line segment joining A ≡ ( x 1 y 1 ) and B ≡ ( x 2 y 1 ) in the ratio ( m : n ) then h = m − n m x 2 − n x 1 & k = m − n m y 2 − n y 1 .
If m = n = 1 then m − n = 0 and P ≡ ( h , k ) becomes undefined . Whereas if m = 1 & n = − 1 then m − n = 2 and P ≡ ( h , k ) becomes the mid point .
Therefore Jim is correct because a point cannot divide externally a line segment in the ratio 1 : 1 as the point will then be undefined. And Jack is wrong because a point can divide externally a line segment in the ratio − 1 : 1 as the point then will be the midpoint.