In the rectangular coordinate system above, if OQ = PQ, is

Question

Untitled.jpg In the rectangular coordinate system above, if OQ >= PQ, is the area of region OPQ less than 30? (1) The coordinates of point P are (6,0). (2) The coordinates of point Q are (3,8)

Bunuel · Answer

https://gmatclub.com/forum/download/file.php?id=21198 In the rectangular coordinate system above, if OQ >= PQ, is the area of region OPQ less than 30? (1) The coordinates of point P are (6,0). We know the length of the base (OP=6) but know nothing about the height (QS), which may be 1 or 100, so the area may or may not be less than 30. Not sufficient. (2) The coordinates of point Q are (3,8). https://gmatclub.com/forum/download/file.php?id=21200 Now, if OQ were equal to PQ then the triangle OPQ would be isosceles and OS would be equal to SP and the area would be: \frac{1}{2}*base*height=\frac{1}{2}(OS+SP)*QS=\frac{1}{2}*(3+3)*8=24 . Since OQ\geq{PQ} then OS\geq{SP} and the base OP is less than or equal to 6, which makes the area less than or equal 24. Sufficient. Answer: B. Similar question to practice: in-the-rectangular-coordinate-system-above-if-op-pq-is-129092.html Hope it's clear. Triangle.png

mikemcgarry · Answer

I'm happy to help. :-)

Statement #1 : The coordinates of point P are (6,0) The base = 6, but the height could be anything, so the area could be anything. This statement, alone and by itself, is not sufficient . Statement #2 : The coordinates of point Q are (3,8) This is tricky. Segment OQ has some length --- we could find this length from the Pythagorean theorem (3^2 + 8^2, then take a square root), but the exact number is not important. The requirement OQ >= PQ allows for two cases --- let's look at them: Case #1 : Suppose OQ = PQ --- then this would be an isosceles triangle, and when PQ came down 8 vertical units from point Q, it would have to go over three horizontal units to (6, 0) --- base = 6, height = 8, area = 12 Case #2 : Suppose OQ > PQ --- then PQ would have to be steeper, and come down the 8 vertical units while going fewer than 3 horizontal units to the right. This would make the base less than 6, so the area would be less than 12. Either way, we have a definitive answer to the prompt. This statement, alone and by itself, is sufficient . Answer = (B) Does this make sense? Mike :-)

PrateekDua · Answer

Thanks Bunuel and Mike, the explanation you provided was way easier than the one I read.

Kudos to both of you!

vipulgoel · Answer

I guess , the coordinates of point O is (0,0) should be given in the stem, unless otherwise could be stated

adityadon · Answer

Hi Mike ,

I could not understand ur explanation for statement B. 
Are we assuming here points O and P are fixed , and we moving point Q in order to consider scenario of OQ>PQ ??
and how can be assume point P to be fixed ?? O is fixed since it is origin

GMATinsight · Answer

Hi Adityadon,

Statement 2: The coordinates of point Q are (3,8)

i.e. X co-ordinate of Q is 3 and Y-co-ordinate is 8 i.e. OQ=  \sqrt{8^2 + 3^2} = \sqrt{73}

BUT Since, OQ >= PQ i.e. PQ  <  \sqrt{73}  But in Triangle PQS since QS=8 therefore PS < 3

i.e. OP = OS + PS = 3 + ( < 3) i.e. OP  < 6

i.e. Area of Triangle OPQ = (1/2)*OP*QS = (1/2)*( < 6)*(8)

i.e.   Area of Triangle OPQ  <  24  Which answers the Questions with Certainty

Hence,   SUFFICIENT

kanigmat011 · Answer

Thanks Mike and Bunuel

bumpbot · Answer

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In the rectangular coordinate system above, if OQ >= PQ, is

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