In the figure above, the point on segment PQ that is twice as far from

In the figure above, the point on segment PQ that is twice as far from P as from Q is

(A) (3,1)
(B) (2,1)
(C) (2,-1)
(D) (1.5,0.5)
(E) (1,0)

Options A or C cannot be correct answer since these points aren't even on segment PQ. E (1,0) is clearly closer to P, so it's also out, D is right in the middle of the segment, so only option B is left.

Answer: B.
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Lets see the algebraic solution.

A point P that divides a line AB internally into ratio m1 & m2 is as below.


A(x1,y1)____________m1_____________P(x3,y3)________________________m2__________________________B(x2,y2)

The formula for finding coordinates are as below.

\(x3 = (x1*m2 + x2*m1)/(m1 + m2)\)

\(y3 = (y1*m2 + y2*m1)/(m1 + m2)\)

Back to our question.

P (0,-1 ) & Q (3,2)

We fathom that m1 =2 & m2 = 1

\(x1 = 0 , y1 = -1\)

\(x2 = 3 , y1 = 2\)

Plugging values, we obtain answer as (2,1). Hence B

Once you know the formula its very easy.

P.S. Bunuel's approach is beautifully elegant. However, for me solving algebraically is much faster than figuring the answer choices out.

The question ask to divide the line PQ into 2:1 ratio and find the point.
By symmetry, the line segment at x-axis (1,0) will be divided in ratio 1:2.
Similarly, at (2,1) line will be divided in ration 2:1

Hence B

Please suggest a algebraic approach, Bunuel!
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I would never solve this question algebraically, but you can check for the tools for that here: math-coordinate-geometry-87652.html
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I was thinking along those lines by forming a triangle and then solving it. This is a fairly easy question you can still use the answer choices but if its a bit more complicated I would like a fast approach.

I think with questions like these, the test writers are testing whether you'd quickly jump to using an algebraic approach, which in this case is much more time consuming, as compared to making the answer choices a part of your toolbox for finding the correct answer..
The question itself tells us we need to split the line into 3 equal parts with the asked coordinate being 2 parts away from P
A quick glance at the graph gives us the slope of 1, which easily shows us which points will cover the three segments
P(0,-1) --> (1,0) --> (2,1) --> Q(3,2)
Thus (2,1) being twice as far from P as from Q

True, an algebraic approach might be required for more complex problems where the slope isn't easily determined or the line segment might be split into a different ratio, but this question isn't testing that

Coordinates of point P = (0,-1)
Coordinates of point Q = (3,2)
Point required to be found is on segment PQ that is twice as far from P as from Q
So, adding (2,2) to point P (0,-1)
Answer = B = (2,1)

Solution:

This is an interesting problem because, although we usually prefer an algebraic approach to solving GMAT questions, the quickest way to solve this problem is to analyze each answer choice in relation to the graph. We need to determine which answer choice gives us a point on the graph that is twice as far from P as from Q. This means that it is closer to Q than to P.

We are given that point Q is at (3,2) and point P is at (0,-1). Let’s start with answer choice A.

A) (3,1)

Looking at the graph we see that (3,1) is not even on line segment PQ. Answer choice A is not correct.

B) (2,1)

Looking at the graph we see that (2,1) is on line segment PQ and it is closer to Q than it is to P. We could use the distance formula to determine the actual distances, but let's wait to see if this is necessary. Let’s test the other answer choices to be certain that answer choice B is correct.

C) (2,-1)

Looking at the graph we see that (2,-1) is not even on line segment PQ. Answer choice C is not correct.

D) (1.5,0.5)

Looking at the graph we see that (1.5,0.5) is on line segment PQ; however, it appears to be about halfway between P and Q. Answer choice D is not correct.

E) (1,0)

Looking at the graph we see that (1,0) is on line segment PQ; however, it is closer to P than to Q. Answer choice E is not correct.
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I felt the question is confusing. Shouldn't it be "In the figure above, the point on segment PQ that is half as far from P as from Q is"?

I think that you're reading it as a comparison between these two things:

- the distance between P and the new point
- the distance between P and Q

However, the wording of the problem is actually asking you to compare these two things:

- the distance between P and the new point
- the distance between the new point and Q
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Do we need to assume that figures are drawn to scale in all the problems? Because I could answer this by just looking at the graph in few seconds.

Problem Solving
Figures: All figures accompanying problem solving questions are intended to provide information useful in solving the problems. Figures are drawn as accurately as possible. Exceptions will be clearly noted. Lines shown as straight are straight, and lines that appear jagged are also straight. The positions of points, angles, regions, etc., exist in the order shown, and angle measures are greater than zero. All figures lie in a plane unless otherwise indicated.

Data Sufficiency:
Figures:
• Figures conform to the information given in the question, but will not necessarily conform to the additional information given in statements (1) and (2).
• Lines shown as straight are straight, and lines that appear jagged are also straight.
• The positions of points, angles, regions, etc., exist in the order shown, and angle measures are greater than zero.
• All figures lie in a plane unless otherwise indicated.
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We can use section formula,

Try this:

https://brilliant.org/wiki/section-formula/

Though I approached the problem similar to Bunuel approach considering the time, The alternative approach can be:

JeffTargetTestPrep if you want to check whether our answer is correct please could you show how we could use the distance formula?

the line is 3 units long. divide in 2 portions as per question
go down one unit on the line from top point and you can locate the point.
The points coordinates can then be seen as (2,1)

The problem can be solved using triangles the lower right-angled triangle is a 45/45/90 triangle therefore the distance from p(0,-1) to x-intercept point (1,0) is √ 2.
The distance from (1,0) to (3,2) is 2√2 therefore the total distance between p and q is 3√2.
Also, the point will be √2 distance along PQ towards Q.
We can imagine a triangle that has a hypotenuse length √2 and sides will be 1 and 1 each.
that means we need one more displacement along with from 1 on X-axis and one displacement from 0 on the Y-axis.
The coordinate of the vertex of this triangle will be ( 2,1)
our answer.

we can do this using weighted averages. But the weights have to be inverted since distance from P is 2 times distance from Q but the point lies closer to Q. So we should take the weights as 1:2 and calculate the point which is pretty straight forward after that.