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Coordinate Geometry

by Darya_NYC » Sun Sep 25, 2011 6:43 pm
I read the explanation of this problem in the book (The Princeton review) and did not understand the explanation. Hope, that someone can help me out.
It is on the coordinate plane. Which of the following is the point on segment XY that is twice as far from X as from Y. Y is (3, root from 3) and X is (0,0)
a. (1,2)
b. (2,1)
c. (1, root from 3)
d. (2, 2 roots from 3)
e. (2,4)

I hope someone can help me :) thanks
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by Darya_NYC » Sun Sep 25, 2011 6:45 pm
or Y might be (3, 3 roots from 3) In the book it says root from 4, but its clearly a mistake, because explanation is talking about roots from 3
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by knight247 » Sun Sep 25, 2011 9:10 pm
Hi Darya,
This problem can be solved using the Section Formula. Please read the following link

https://www.learnnext.com/class10/maths/ ... ormula.htm

It gives the details of the Section Formula. Its pretty easy. I will post the solution to the problem in a while.
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by knight247 » Wed Sep 28, 2011 11:27 am
Look at the image I have uploaded to understand the section formula.
Now, suppose we have a line AB and the co-ordinates of A are (x1,y1) and of B are (x2,y2). Suppose there is a point O on the line that divides the line in the ration M:N then the co-ordinates of that point can be found out by using the Section Formula i.e.

X co-ordinate of point O=(M*x2+N*x1)/(m+n)
Y co-ordinate of point O=(M*y2+N*y1)/(m+n)
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by knight247 » Wed Sep 28, 2011 11:42 am
Now, coming back to the problem. Look at the diagram that i've attached. So basically we have a point on the line XY that is twice as far from X as from Y i.e. this point divides line XY in the ratio 2:1

So, now using Section Formula we have,
x1=0,y1=0
x2=3, y2=√3
m=2 and n=1

So, the x-co-ordinate of O=(2*3+1*0)/3=6/3=2
the y-co-ordinate of O=(2*√3+1*0)/3=2√3/3=2/√3

So the co-ordinates of the required point are (2,2/√3) Which doesn't seem to be in the Answer choices. Can you check the answer choices again. And, hope i've not made any silly mistakes.lol
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by GMATGuruNY » Thu Sep 29, 2011 1:39 am
Darya_NYC wrote:I read the explanation of this problem in the book (The Princeton review) and did not understand the explanation. Hope, that someone can help me out.
It is on the coordinate plane. Which of the following is the point on segment XY that is twice as far from X as from Y. Y is (3, root from 3) and X is (0,0)
a. (1,2)
b. (2,1)
c. (1, root from 3)
d. (2, 2 roots from 3)
e. (2,4)

I hope someone can help me :) thanks
In a 30-60-90 triangle, the sides are proportioned x : x√3 : 2x.
The DISTANCE between two points = the HYPOTENUSE of a right triangle:

Image

The drawing above shows that the legs of the larger triangle are 3 and 3√3, indicating that the larger triangle is a 30-60-90 triangle with sides of 3, 3√3 and 6.
The smaller triangle is a 30-60-90 triangle with sides of 2, 2√3 and 4.
Since Z is 4 units from X and 2 units from Y, Z is twice as far from X as from Y.
The coordinates of Z are (2,2√3).

The correct answer is D.
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by Darya_NYC » Thu Sep 29, 2011 5:28 pm
Thank you, guys! I think I got it :)
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