Is (-1,-1) a point on line OQ?

Question

OQ.png Is (-1,-1) a point on line OQ? (1) P = (-2,2) (2) Angle POQ = 90 degrees

Bunuel · Accepted Answer

https://gmatclub.com/forum/download/file.php?id=18493 
 Is (-1,-1) a point on line OQ?

(1) P = (-2,2). This implies that point P lies on line y = -x. No info about line segment OQ. Not sufficient.

(2) Angle POQ = 90 degrees. We can rotate angle POQ so that (-1,-1) to lie as well as not to lie on OQ. Not sufficient.

(1)+(2) Since P lies on line y = -x and angle POQ = 90 degrees, then point Q (line segment OQ) lies on line y=x. Therefore, point (-1,-1) lies on line OQ. Sufficient.

Answer: C.

Hope it's clear.

1022lapog · Answer

Hi Bunnel, when you say "rotate angle POQ" what do you mean? the quadrant that these two lines lie in the figure is also not implicit in GMAT unless specified? I know the scale is not implicit unless specified. But this figure clearly shows that the lines are in quadrants (-,+) and (-,-). So in that case b alone is sufficient to answer this question. right?

Bunuel · Answer

P must be in the second quadrant and Q must be in the third quadrant. But you can rotate within them. Consider the diagram below: Rotate.png So, as you can see, the second statement alone is not sufficient. Hope it's clear.

teal · Answer

(1)+(2) Since P lies on line y=x and angle POQ = 90 degrees, then point Q (line segment OQ) lies on line y=x. Therefore, point (-1,-1) lies on line OQ. Sufficient.

hello Bunuel,

I don't understand the above explanation when you combine the two statements, P Lies on y =-x and angle POQ =90 then how do we conclude that point Q lies on y=x. Do P and Q have the value of x coordinate ? Is that info provided from figure?

Bunuel · Answer

Lines y=x and y=-x are perpendicular to each other and both pass through the origin: Untitled.png O P is part of y=-x (notice that O is the origin), thus O Q must be part of y=x.

closed271 · Answer

(1) Nothing about the co-ordinates Q can be stated with certainty.
(2) This tells that angle POQ is 90 deg. Nothing concrete about Q.

(1) + (2) If a line joining P and Q is drawn, then a right angled triangle POQ forms. OP can be calculated. Assuming Q lies on the line OQ, distance OQ can be calculated. Then the hypotenuse can calculated. If it satisfies Pythagorean theorem, then (-1,-1) lies on OQ, else it does not. Hence, (C). Note that there is no need to compute any of the lengths. We just need to think whether the the lengths can be computed. If they can (which is the case here), then (-1,-1) may or may not be a point on OQ, hence (C).

LogicGuru1 · Answer

Statement1) Gives us slope of the line passing through PQ as -1. Says nothing about line OQ
Insufficient

Statement 2) tells us that slope of line PQ will be a inverse reciprocal of the line OQ, but does not give any info about slope 
Insufficient

Merge both statements

PQ's slope = -1
OQ slope = 1
Hence OQ will definitely pass from (-1,-1)

ac556 · Answer

Hi Bunnel, In case the angle POQ was given to be anything other than 90 degrees then what would be the answer?

Bunuel · Answer

The answer would still be C but the answer to the question would be NO, point (-1,-1) doe not lie on line OQ.

ac556 · Answer

Hi Bunnel, Could you please explain as to how we will be able to tell that the point will not lie on OQ if the angle was say 60 degrees. Thanks

Bunuel · Answer

No matter what is the measure of angle POQ, line segment OQ would be fixed (because P is fixed the measure of angle POQ will fix the line segment OQ), so we would be able to answer whether it contains point (-1,-1) . Also, the only way it to contain (-1,-1) is when OQ is on y=x, or in other words when POQ is 90 degrees.

siddyj94 · Answer

Bunuel  - Please clear my thought process as when we combine the statements it states that  P=(-2,-2)  and  angle POQ is 90 deg

1) Can we assume from this and the figure that the two points are figuratively on the opposite ends as they lie between y=x and y=-x and can jump on the conclusion that OQ does carry the point (-1,-1).

2) Secondly not able to figure out the meaning of the rotation within them?

Thanks in advance.

Bunuel · Answer

When we combine the statements we can infer that P is in the II quadrant and somewhere on line y = -x. Similarly we can infer Q is in the III quadrant and somewhere on line y = x.

kaleem765 · Answer

Statement 1: Only tells the location of point P. Insufficient
Statement 2: Only tells that both lines are perpendicular to each other at origin (0,0)

Combining statement 1&2: we know that lines are perpendicular to each other so the slope of one line will be the negative reciprocal of the other line. Using point P(-2,2) and O(0,0), the slope of line OP is -1. so the slope of line OQ will be the negative reciprocal of -1 i.e. 1.

If the slope, using points (-1,-1) and (0,0), is also 1, then it means point (-1,-1) lies on line OQ.

using the slope formula and using points (-1,-1) and (0,0): m = (-1-0)/(-1-0) = 1

so point (-1,-1) lies on line OQ.

Answer: C

ak2121 · Answer

Just wanted to know that those small lines on line segment op and oq don't mean that they equal?

Bunuel · Answer

Those lines meant to be dots I think but whether OP = OQ is actually irrelevant.

ak2121 · Answer

If they are equal, won't it mean that oq is (-2,-2)?

Bunuel · Answer

You mean from (1)? No, if we were told that OP = OQ and P = (-2,2), then it would not necessarily mean that Q = (-2, -2). We don't know the measure of POQ - if it's 90 degrees then yes, Q = (-2, -2) but if not, then Q will not be at (-2, -2). But again the lengths of OP and OQ are absolutely irrelevant here. We re asked to find whether (-1,-1) is a point on line OQ, so we need to know the slope of line OQ, not the length of line segment OQ.

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Is (-1,-1) a point on line OQ?

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