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What are the coordinates for the point on Line AB

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What are the coordinates for the point on Line AB [#permalink]

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What are the coordinates for the point on Line AB (see figure) that is three times as far from A as from B, and that is in between points A and B?
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OA:
[Reveal] Spoiler:
(-2.75, 1.5)

Originally posted by hoaivinh on 10 Sep 2011, 08:26.
Last edited by Bunuel on 21 Aug 2013, 14:41, edited 2 times in total.
Edited the question, added the figure and the OA.
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Re: Coordinate Plane - Find the point with distance [#permalink]

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New post 10 Sep 2011, 09:39
hoaivinh wrote:
Hi folks,

This is a rather hard question on coordinate plane and I need your help.
"What are the coordinates for the point on line AB that is 3 times as far as from B, and that is between points A and B, knowing that A(-5,6) and B(-2,0)"

At first, I worked out the x-axis distance from A to B is 3 (-5+2) and y-axis distance from A to B is 6 (6-0). I divide 4 segments, x for per segments, then I have 4x. 3x from A and x from B. Then I get stuck there. But I do not understand why the explanation in the answer is that "3x=4, x=0.75. The point is located on x-axis -2.75".
I really do not understand.

Thank you.

The x-coordinate of the point is x=0.75 away from -2, and thus it's located on -2.75
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Re: Coordinate Plane - Find the point with distance [#permalink]

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Silver89 wrote:
hoaivinh wrote:
Hi folks,

This is a rather hard question on coordinate plane and I need your help.
"What are the coordinates for the point on line AB that is 3 times as far as from B, and that is between points A and B, knowing that A(-5,6) and B(-2,0)"

At first, I worked out the x-axis distance from A to B is 3 (-5+2) and y-axis distance from A to B is 6 (6-0). I divide 4 segments, x for per segments, then I have 4x. 3x from A and x from B. Then I get stuck there. But I do not understand why the explanation in the answer is that "3x=4, x=0.75. The point is located on x-axis -2.75".
I really do not understand.

Thank you.

The x-coordinate of the point is x=0.75 away from -2, and thus it's located on -2.75


Thank you Silver89. But how is 3x=4? That's the thing I got stuck.
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Re: What are the coordinates for the point on line AB that is 3 [#permalink]

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New post 21 Aug 2013, 13:34
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Well, in this case, I think it is a typo. Because they say 3x=4 is equal to x=0.75 which is not true because x = 4/3 ----or 1.25 in decimal.

They associate 4x=3 which is the difference between -5 and -3. So x=0.75 to which they add the -2 from the X-coordinates, which gives you -2.75.

You do the same for the y-axis 4x=6 which gives x=1.5.

The answer is -2.75,1.5
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Re: What are the coordinates for the point on line AB that is 3 [#permalink]

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New post 21 Aug 2013, 14:42
gabbag00 wrote:
Well, in this case, I think it is a typo. Because they say 3x=4 is equal to x=0.75 which is not true because x = 4/3 ----or 1.25 in decimal.

They associate 4x=3 which is the difference between -5 and -3. So x=0.75 to which they add the -2 from the X-coordinates, which gives you -2.75.

You do the same for the y-axis 4x=6 which gives x=1.5.

The answer is -2.75,1.5


Edited the question, added the figure and the OA: what-are-the-coordinates-for-the-point-on-line-ab-120415.html#p973190
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Re: What are the coordinates for the point on Line AB [#permalink]

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Refer the formula attached in the figure below.

Let P (x,y) be a point on line AB such that AP/PB = 3/1.

Co-ordinate of X = \(\frac{mx2 + nx1}{m + n}\) -----> \(\frac{3(-2) + 1(-5)}{3+1}\) ----------> \(\frac{-6 - 5}{4}\) --------> \(\frac{-11}{4}\)

Co-ordinate of Y = \(\frac{my2 + ny1}{m + n}\) -----> \(\frac{3(0) + 1(6)}{3+1}\) ----------> \(\frac{6}{4}\) -------> \(\frac{3}{2}\)

Hence p(x,y) = \((\frac{-11}{4})\), \((\frac{3}{2})\)

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Re: What are the coordinates for the point on Line AB [#permalink]

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vishalrastogi wrote:
What are the coordinates for the points on line AB (see figure) that is three times as far from A as from B, and that is in between points A and B?


Please post the OA next time.

Since the point lies on the line AB it should satisfy the line equation: y+6 = (0-6)/(-2+5)*(x+5)
Therefore, y+6 = -2(x+5)

The only option that satisfies this equation is: X = -2.75, y = 1.5
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What are the coordinates for the point on Line AB [#permalink]

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New post 28 Nov 2014, 01:47
Hi experts, I found this approach in a flashcard to this question:
a) The total x-axis distance from one point to the other is -3 (-5 - -2)
b) The total y-axis distance from one point to other is 6 (6 – 0)
c) Considering that the distance between A and B is 4x, and we need a point which is 3x distant from A and x distance from B.
d) 4x = -3 --> x = -0.75. (Have corrected the typo here seeing a post above) The point is located on x-axis at -2.75
e) 4y = 6 --> y = 1.5. The point is located on y-axis at 1.5
f) The answer is (-2.75, 1.5)

Kindly help me understand how the 0.75 becomes -2.75 and how the 1.5 remains as 1.5.
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Re: What are the coordinates for the point on Line AB [#permalink]

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So for this question couldn't we just take the midpoint between A and B and then take that answer and find the midpoint between midpoint of AB and B?

Gives us -11/4, 3/2
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Re: What are the coordinates for the point on Line AB [#permalink]

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Hi! See attached. I have worked it out in ppt and attached it in this document.
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Re: What are the coordinates for the point on Line AB [#permalink]

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New post 22 Jun 2016, 03:47
Can somebody please tell me where I am going wrong.
Taking ABC with A being ( -5,6) , B being (-2,0) and C being (-5,0) . Length of AC is 6 cm. Length of CB is 3 cm .
Using Pytho theorem we can get that -

6^2 + 3^2 = AB^2
36 + 9 = 45 = AB ^ 2

So, AB = 4X =
X = sqrt(45) /4

Now using similiarity , we can find that the x and y co-ordinates of the smaller triangle.
Mark the small triangle as BED .

ED / 6 = [ sqrt(45)/4 ] / sqrt(45) = 1/4
ED or the Y co-ordinate for the point D will be = 6/4 = 1.5


EB / 3 = 1/4
EB or X co-ordinate for the point D will be = 3/4 = 0.75

I know I am wrong somewhere ! Can somebody please point it out to clear my concept.
Thanks in advance :)
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Re: What are the coordinates for the point on Line AB [#permalink]

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hoaivinh wrote:
What are the coordinates for the point on Line AB (see figure) that is three times as far from A as from B, and that is in between points A and B?
Attachment:
The attachment mgmat.png is no longer available


OA:
[Reveal] Spoiler:
(-2.75, 1.5)


It is critical to know the internal section formula in order to solve this question. Basically, this formula allows us to solve for a point on a line such as line AB , let's say point "P" , provided we know the ratio of the distance between that point and the other two end points on that line ( the end points are the coordinates given in the problem ) which is this is scenario AP to BP. Now, there's two ways of understanding the formula- what the question is referring to when it states a point that is "three times as far from A as from B" is that if you split line AB into two parts then the length between A and the hypothetical (imaginary point in question) P is 3 times greater than P to point B. Algebraically, we can say this ratio is 3x to x, or really just more simply 3 to 1. But as we look at the diagram how do we really understand each component of the internal section formula? That is to say how do we appropriately plug in 3 and 1 to m and n. Should 3 be m or should 3 be n? If we reverse engineer the formula

[{m(x2) + n(x1) / m + n} , {m(y2) + n (y1)/ m +n}] - the sum of our ratio, m and n, equals the denominator of the fraction.

But wait this is also the same thing as

m/ m + n (x2, y2) + n/m +n (x1,y2)

The key to properly assigning each x coordinate to either m or n respectively depends on where you partition the line- this is why a triangle helps. So if we take coordinate A for example such as in the both of my diagrams and the problem, we would align the coordinate of A with the number that corresponds to ratio of the line that is across the partition.
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What are the coordinates for the point on Line AB [#permalink]

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hoaivinh wrote:
What are the coordinates for the point on Line AB (see figure) that is three times as far from A as from B, and that is in between points A and B?
Attachment:
mgmat.png


OA:
[Reveal] Spoiler:
(-2.75, 1.5)


the points are A(-5,6) and B (-2,0)
We need to find the point which divides AB in the ratio 3:1

It is given by the formula (mx2 +nx1)/(m+n) , (my2+ny1)/(m+n)

So, [(3*-2+1*-5)/(3+1) , (3*0+1*6)/(3+1)] = [-11/4, 3/2] is the required point...
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Originally posted by shashankism on 15 Jun 2017, 07:12.
Last edited by shashankism on 15 Jun 2017, 23:18, edited 1 time in total.
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What are the coordinates for the point on Line AB [#permalink]

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New post 15 Jun 2017, 18:59
hoaivinh wrote:
What are the coordinates for the point on Line AB (see figure) that is three times as far from A as from B, and that is in between points A and B?
Attachment:
The attachment mgmat.png is no longer available


OA:
[Reveal] Spoiler:
(-2.75, 1.5)


Here is an updated diagram- hopefully the color coding helps make it more clear how to correctly plug in each set of coordinates with correct ratio- here we ca see m is 2 and therefore it must be matched up with the coordinate pair as per the internal section formula

[2/2+ 3 (7,9)] , [ 3/2 +3 (5,3)] = m/ m + n (7,9) , n/ m + n (5,3) = P(x,y) - here p is the imaginary point between A and B

OR

mx2x1/ m + n , my2y1/ m + n = P(x,y) - mathematically this is the same as the formula but just compressed- here's why

2 (7) / (2+3) + 3 (5)/ 2 +3) =
14/ (2 +3) + 15/ (2+3) is the same thing as 14 + 15/ 2+ 3 - and when you add these you get the x coordinate so instead of multiplying it as through the first formula the second formula just compresses everything- but for the sake of understanding the concept the first formula may be better to start with.
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What are the coordinates for the point on Line AB [#permalink]

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New post 15 Jun 2017, 23:20
Nunuboy1994 .. Well you have presented the problem with a nice diagram..
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Re: What are the coordinates for the point on Line AB [#permalink]

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New post 17 Jun 2017, 00:13
gmatprep2o16 wrote:
Can somebody please tell me where I am going wrong.
Taking ABC with A being ( -5,6) , B being (-2,0) and C being (-5,0) . Length of AC is 6 cm. Length of CB is 3 cm .
Using Pytho theorem we can get that -

6^2 + 3^2 = AB^2
36 + 9 = 45 = AB ^ 2

So, AB = 4X =
X = sqrt(45) /4

Now using similiarity , we can find that the x and y co-ordinates of the smaller triangle.
Mark the small triangle as BED .

ED / 6 = [ sqrt(45)/4 ] / sqrt(45) = 1/4
ED or the Y co-ordinate for the point D will be = 6/4 = 1.5


EB / 3 = 1/4
EB or X co-ordinate for the point D will be = 3/4 = 0.75

I know I am wrong somewhere ! Can somebody please point it out to clear my concept.
Thanks in advance :)



WHy are you trying to solve this way ..

Something is not clear in your solution. You are just dividing the distance between AB and taking it as coordinate...

Try to use internal section formula.. it is much easier to solve and will save you a lots of time. After all getting a good score in GMAT is all about time. So try to find shorter methods to solve question and move towards next question ASAP. Otherwise getting a 50 or 51 in QA will be difficult.
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What are the coordinates for the point on Line AB [#permalink]

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New post 11 Dec 2017, 11:41
Hi! Where this problem comes from? Not official GMAT? Do real GMAT problems like that appear on the exam so that I need to memorize this internal section formula (which by the way is never included in any theoretical basis for GMAT and even unofficial flashcards (this problem is solved differently not via formula there?)?
I also found it is easier to resolve this question by simple dividing by 4 of x and y distance and adding to existing point... I do not see any need to know this internal section formula here at all..
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Re: What are the coordinates for the point on Line AB [#permalink]

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New post 20 Mar 2018, 14:56
Could Bunuel or chetan2u provide a solution to this problem? Thanks!
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Re: What are the coordinates for the point on Line AB [#permalink]

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mahrah wrote:
Could Bunuel or chetan2u provide a solution to this problem? Thanks!



Hi..
A very easy approach will be to make similar triangles..
What it will mean is ..
Whatever effect it has on the the line given, the SAME effect will be on x and y coordinates..

So the points are closer to B..
1) x coordinate
Weighted average method..
Divide the distance by 4 as per the question and add to coordinate of B..
So -2+(-5-(-2))/4=-2-0.75=-2.75
2) y coordinate..
Similarly y= 0+(6-0)/4=0+1.5=1.5

(X,y)=(-2.75,1.5)
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Re: What are the coordinates for the point on Line AB   [#permalink] 20 Mar 2018, 18:48
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