In the figure above,line segment OP has slope 1/2 and line segment PQ

Lets Assume P (a,b) and Q (x,y)
It is given that
Slope of OP
\(b/a=1/2\)
\(a=2b\)

Slope of OQ is \(y/x\)...Required..

Slope of PQ will be \(y-b/x-a = 2\)

y-b=2(x-2b)
y=2x-3b

(1)
OP = 2\sqrt{5}
OP = \sqrt{(a)^2 + (b^2)}
20 = a^2 + b^2
We already know a =2b
20 = 5b^2
b=+2 or -2
then a would be +4 or -4
If we use (4,2) to solve for (x,y) to reach to slope then
y-2/x-4 =2
y-2=2x-8
y=2x-6 We do not have another equation to solve for (x,y)

If we use (-4,-2) to solve for (x,y) to reach slope then
y+2/x+4 =2
y+2=2x+8
y=2x+6 We do not have another equation to solve for (x,y)

Hence Not Sufficient

(2) Q(5,4)--> x=5, y=4

y/x = 4/5

Bunuel Can u explain the approach, my wife is saying something is wrong

mikemcgarry why are we allowed to assume that O is at origin?

Dear ambar09d,

I'm happy to respond.

My friend, on GMAT PS geometry diagrams, there are lots of special cases that we can't assume---we can assume parallel or right angles or two equal sides simply because something looks that way. "Almost" doesn't count in geometry.

Nevertheless, there are a few basic assumptions we can make. If a line looks straight, we can assume that it is straight, that is has no hidden bend or curve to it. Also, if a point is portrayed as "on" a line, we can assume that it is in fact, "on" the line and not simply near the the line. In this diagram, point O is shown at the juncture of the x- and y-axes. We absolutely can assume that it is on both of those lines, and therefore, must be the intersection, the origin. Also, conventionally, the origin is often labeled O. For a problem to have a point labeled O near the origin but not exactly on it would a level of deviousness that we simply don't see on the GMAT.

Does all this make sense?
Mike

i am confused. i think d is correct.

in 1
we now lengthe and slope we can find the coordinate of p
similarly we can find coordinate of q

and 1 is sufficient.

Dear thangvietnam,

I'm happy to respond.

From Statement #1, we are given the length of OP, and we already know the slope of OP, so we absolutely could find the (x, y) coordinates of point P. The trouble is that this is NOT what the question asks. It's asking for the slope of OQ, and having full information about OP and point P tells us zilch about point Q or segment OQ. From the prompt & Statement #1 combined, we know neither the length nor the slope of OQ--in fact, the question is asking for that slope.

Does all this make sense?
Mike

mikemcgarry
Hi, I'm totally convinced that B is right answer.
Just want to ask further.
What if statement 1 gives the slope of line OP (eg: 2), can statement 1 alone also sufficient?
Thank you so much in advance

Dear lichting,

I'm happy to respond.

From the prompt information, we only know one angle in the triangle, angle P. (We don't explicitly have the value of that angle, but with the slope information and trigonometry--beyond what the GMAT expects you to know, we could find that angle. As it happens, that angle is 143°7'48.368", but you don't need to know how to find that.) The point is, from the fact that the slopes of OP and PQ are determined and fixed, angle P is locked into place at a specific value. All we know from the prompt is that one angle.

As it is, Statement #1 gives us only a length--thus, for triangle OPQ we have one length and one angle, and we can't find anything from that combination.

Now, your question is confusing to me. You see, in any GMAT DS questions, 100% of the time, the information in the prompt alone, without the statements, is always always always insufficient. We can never give a definitive answer to the DS prompt question solely from information in the prompt--the statements may or may not provide sufficient information, but without the additional information in the statements, we never can get anywhere. Thus, any information already present in the prompt never gets us any closer to sufficiency. If the information we have at any point is not sufficient, then repeating any part of it does absolutely zilch for us. This is a very common human misunderstanding that advertising and politicians like to exploit: repeating something doesn't actually make it more true. Thus, repeating the slope of OP, which we already know, does't absolutely zero to enhance the sufficiency of statement #1.

Let's say the question were different, and the prompt were asking for, say, the length of PQ.
Then, if statement #1 told the length of OP and the slope of OQ, then from all three slope, we could find all the angles in the triangle, and with the value of the length, we could find all three sides. When I say "we could find them"--they are mathematical fixed with definite values as a result of that information, although finding the actual numerical lengths and angles would involve using trigonometry--again, beyond what the GMAT expects. Actually, this wouldn't be a real GMAT DS question, because the GMAT is scrupulous about not asking you to determine the sufficiency of quantities that are technically mathematically fixed, although the math to find them is beyond GMAT math: the GMAT DS never goes into that territory.

You may find this blog article relevant:
GMAT Data Sufficiency: Congruence Rules

Does all this make sense?
Mike

No calculations needed. Just think about which points are uniquely identifiable and which are not.

We need slope of OQ so for that we need the co-ordinates of O and Q. We know O is (0, 0) so we need the co-ordinates of Q only.

Stmnt 1: OP has length \(2\sqrt{5}\)

Slope of OP is known so we can draw a line passing through O with slope 1/2. Since length of OP is given, we know where P will lie on that line so P is uniquely defined. Slope of PQ is 2 so we can draw a line passing through P with slope 2. But length of PQ is not known. Do we know then where Q will lie on this line? No. If I move Q a little bit ahead on the line (away from P), slope of OQ will increase. Since Q is not defined, we cannot find the slope of OQ.
Not sufficient.

Stmtn 2: Q is (5, 4)
Now Q is a defined point so we can easily find the slope of OQ.
Sufficient.

Answer (B)

Solution:

We need to determine the slope of line segment OQ, given that OP has slope 1/2 and PQ has slope 2. Notice that if we know or can determine the coordinates of point Q, then we can determine the slope of OQ since point O is the origin.

Statement One Alone:

From the information, we can determine the coordinates of point P. However, even if we can determine coordinates of point P, we can’t determine the coordinates of point Q since there are many possible coordinates for Q such that PQ has slope 2. Statement one alone is not sufficient.

Statement Two Alone:

Since we know the coordinates of Q, we see that the slope of OQ is (4 - 0) / (5 - 0) = 4/5. Statement two alone is sufficient.

Answer: B

Hi KarishmaB



How were you so confidently able to skip the inference part? I know that we need to find the co-ordinates of Q to find the slope of OQ. However, even then I tried to infer as much as I can from the question stem first which took some time.

I am not sure what you mean by the 'inference part.'
I can find the slope of OQ if I know the line on which OQ lies or the co-ordinates of point Q itself (since I will know 2 points on that line then, O and Q and can find the equation, and hence the slope, of the line).
But I wouldn't waste a lot of time in pre-thinking since the statements will give me additional data.

For DS Geometry questions, I often try to make a diagram in my head and see what info I need to make a unique, defined diagram.

Let me explain that with statement 1 info.

We have the following data to use with Statement 1 alone: Slope of OP = 1/2, Slope of PQ = 2, Length of OP = 2*sqrt5

I will imagine the co-ordinate axis on the graph. Passing through O, every slope gives a defined line (infinitely extending at both ends) on the axis.

Each line passing through the centre in the figure below has a distinct slope:


Given a point and a slope, I can draw a unique line.

So I know I get a distinct line with slope 1/2 passing through O so I draw that (the line will pass through (2,1), (4,2) (6,3) etc).
Since length OP (= 2*sqrt5) is given, I will measure that on my compass and cut off the line to get the point P.
Through P, I uniquely draw a line with slope 2 which extends infinitely on both sides. Now Q lies on this line, but there? I can move the point Q on this line and in every case, OQ will give me a different slope.
This means statement 1 alone is not sufficient.

Try to construct this using a paper pencil and see if it makes sense.

I chose E because of origin assumption. But X & Y axis were pretty straight give away