SOLUTION: Find the coordinates of P such that P is a point on AB, A(0,6), B(6,0), AP:AB=2:1

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Question 1174279: Find the coordinates of P such that P is a point on AB, A(0,6), B(6,0), AP:AB=2:1



Found 4 solutions by greenestamps, Edwin McCravy, ikleyn, AnlytcPhil:
Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!


Work with the x and y coordinates separately, separating each of the differences into two parts in the ratio 2:1.

x: A to B is a change of +6, from 0 to 6; divide it into +4 and +2. The x coordinate of P is 0+4 = 4.

y: A to B is a change of -6, from 6 to 0; divide it into -4 and -2. The y coordinate of P is 6+(-4) = 2.

ANSWER: P(4,2)

------------------------------------------------

Instead of solving the problem using ratios, here is a different way to look at solving the problem, using fractions.

The x coordinate of P is 2/3 of the way from A to B -- that is, 2/3 of the way from 0 to 6. A to B is a change of 6; 2/3 of 6 is 4; the x coordinate of P is 0+4 = 4.

The y coordinate of P is 2/3 of the way from A to B -- that is, 2/3 of the way from 6 to 0. A to B is a change of -6; 2/3 of -6 is -4; the y coordinate of P is6+(-4) = 2.
Answer by Edwin McCravy(20056)   (Show Source): You can put this solution on YOUR website!
Greenstamps misread the problem.  In order for AP:AB = 2:1, P is just the
double extension of AB down to P(12,-6). P is on the infinitely long
line . P could not be on the line segment (of finite length). . 
 
 

I just realized there is another solution (-12,18)



Edwin
Answer by ikleyn(52793)   (Show Source): You can put this solution on YOUR website!
.
Find the coordinates of P such that P is a point on AB, A(0,6), B(6,0), AP:AB=2:1
~~~~~~~~~~~~~~~~~~~


As the problem is worded,  printed,  posted and presented,  IT  HAS  NO  solution.

It can not be a point  P   "on AB"   such that   AP : AB = 2:1.

Such a point  P  is  NECESSARY  out of the segment  AB.


-------------

After reading the post by Edwin. 

    Edwin, I do not see how your thoughts do relate to my post.


    I only know, that if a person, who positions himself (or herself) as a Math problem composer,

    has enough (=adequate) Mathematical culture, he (or she) always can (should, MUST) formulate Math problem

    in a precisely accurate way, excluding any ambiguities.


    It is the form of Mathematical politeness and mathematical maturity of such a Math composer.


        +-----------------------------------------------------------------+
        |     In the context of this problem, as it is worded, printed    |
        |     posted and presented, the objects denoted by two capital    | 
        |        letters, such as AB, AP, BP, represent SEGMENTS          |
        |           and DO NOT REPRESENT lines or intervals.              |
        +-----------------------------------------------------------------+

    
    THEREFOTRE, I consider your reproaches to me as INCONSISTENT,

            and expect to get apologies from you.


    Also, please notice that different parts of your comment,
    in the context of the given problem, CONTRADICT to each other.

In this forum, there are many people among the tutors, who are agitated to correct me or to teach me,
instead of learning from me.



Answer by AnlytcPhil(1806)   (Show Source): You can put this solution on YOUR website!
Ikleyn is not aware of the inconsistencies in geometry notation. Copied and pasted from:
 
https://www.mathstips.com/point-ray-line-and-line-segment/

A line is named by any two points on it and written as line AB or line PQ.
One and only one line can be drawn passing through two given points A and B.
This line is called AB. It may also be called BA.  Line BA is the same as line
AB. Both pass through the same two points A and B.

In other places, the notation is different.  Often line segments are called "lines".

In my many years of teaching mathematics, I have discovered that notation in
mathematics is very often inconsistent.  Also some books use "equal" for lines
and angles, and "congruent" only for closed figures, whereas others use
"congruent" for them all.  Nothing is written in stone.

As another example of inconsistency, does (1,2) represent a point or an open
interval?  We can only go by context.  

Edwin
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