What are the coordinates for the point on Line AB that is

Question

https://gmatclub.com/forum/download/file.php?id=20837 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? A. (-2.75, 1.5) B. (-2.65, 1.5) C. (-2.55, 1.5) D. (-2.75, 2.5) E. (-2.75, 3.5) mgmat.png

Narenn · Accepted Answer

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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Hope that helps!

Attachments

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lalania1 · Answer

Hi! See attached. I have worked it out in ppt and attached it in this document.

Silver89 · Answer

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

hoaivinh · Answer

Thank you Silver89. But how is 3x=4? That's the thing I got stuck.

gabbag00 · Answer

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

Bunuel · Answer

Edited the question, added the figure and the OA: what-are-the-coordinates-for-the-point-on-line-ab-120415.html#p973190

igotthis · Answer

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

AdmitJA · Answer

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.

10046844 · Answer

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

gmatprep2o16 · Answer

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

Nunuboy1994 · Answer

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

- 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.

shashankism · Answer

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,   =   is the required point...

Nunuboy1994 · Answer

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

,   = 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.

shashankism · Answer

Nunuboy1994  .. Well you have presented the problem with a nice diagram..

shashankism · Answer

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.

Erjan_S · Answer

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..

mahrah · Answer

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

chetan2u · Answer

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)

abuzafor · Answer

Narenn!

For clear about section formula( external ), Give an example.

What are the coordinates for the point on Line AB that is

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What are the coordinates for the point on Line AB that is

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