Problem Solving

chuotsmart wrote:OG 12 edition, DS section 149.



In the rectangular coordinate system above, if OP < PQ, is the area of region OPQ greater than 48 ?

(1) The coordinates of point P are (6,8).
(2) The coordinates of point Q are (13,0).

(1) is Sufficient.

(2) is Insufficient, why? Because with OQ = 13, I can prove that the area of the triangle is smaller than 48.

My solution:

Triangle OPQ's area = (1/2) * bh, suppose PH is the height of the triangle, the area of the triangle OPQ = (1/2) * (PH * OQ)

From the (2) statement, we have OQ =13 and the area of the triangle OPQ = 1/2 * (13* PH) = 6.5 * PH

Triangle OPQ lies in a coordinate system. Hence, OQ - OP = PQ => OQ = PQ + OP +> 13 = PQ + OP. We also know that OP < PQ, so OP < 13/2 (S1)

In the tringular OHP, PH is one of the legs and OP is the hypotenuse, so PH < OP (S2)

(S1) + (S2): PH < 6.5, so 6.5*PH < 6.5*6.5.

6.5*6.5 ~ 42 < 48. So the area of OPQ is NOT greater than 48.

The second statement is sufficient

You might want to check the part highlighted in red.
What you've wrote is only possible if O, P and Q are collinear points. But that may not be the case.

@ anurag

Anurag@Gurome wrote:You might want to check the part highlighted in red.
What you've wrote is only possible if O, P and Q are collinear points. But that may not be the case.

yep
in fact, you should say "can't", not just "may not". those 3 points form a triangle, so they can't be collinear.

--

@ original poster
that's ... a lot of theory.

to realize that statement 2 is insufficient, just consider the fact that you have a triangle whose base goes from (0, 0) to (13, 0), and then a top vertex P that can be literally anywhere above that base.
i.e., P could be at, say, (5, 0.00001), in which case the area of the triangle is way less than 48. or P could be at, say, (5, 1,000,000), in which case the area of the triangle is much, much greater than 48.

--

finally, on this problem, if you know the conventions by which gmac creates data sufficiency problems, you can actually figure out that the second statement CAN'T mean what you think it means.
here's how:
on an official DS problem, the two numbered statements will NEVER contradict each other.
if you think that the two numbered statements are mutually contradictory, your interpretation of at least one of them is definitely wrong.

in this problem, the first statement guarantees that the area of the triangle is greater than 48; you probably understand this, since you've posted as if it's no problem.
therefore, if your interpretation of the second statement leads you to conclude that the area must be less than 48, then that interpretation must be wrong: you can't have one statement that guarantees that the area is less than 48, but another one that guarantees that the area is more than 48.

lunarpower wrote:@ anurag

Anurag@Gurome wrote:You might want to check the part highlighted in red.
What you've wrote is only possible if O, P and Q are collinear points. But that may not be the case.

yep
in fact, you should say "can't", not just "may not". those 3 points form a triangle, so they can't be collinear.

Thanks Ron.
But the question never says that the points O, P and Q forms a triangle. It only says "In the rectangular coordinate system above, if OP < PQ...". I think without any other information we cannot say whether they are collinear or not.

Anurag@Gurome wrote:

lunarpower wrote:@ anurag

Anurag@Gurome wrote:You might want to check the part highlighted in red.
What you've wrote is only possible if O, P and Q are collinear points. But that may not be the case.

yep
in fact, you should say "can't", not just "may not". those 3 points form a triangle, so they can't be collinear.

Thanks Ron.
But the question never says that the points O, P and Q forms a triangle. It only says "In the rectangular coordinate system above, if OP < PQ...". I think without any other information we cannot say whether they are collinear or not.

in the original source, the problem is accompanied by a diagram, which shows o, p, and q as the three vertices of a triangle. the original poster has reproduced that diagram (although the added altitude and point H are not part of the original diagram).

tyronetan82 wrote:Ron,

Need your help to clarify. O in the diagram seem to be at the origin (0,0) of the coordinate system. But I thought the rule on the GMAT was never to trust the diagram unless facts are clearly stated, and so nothing on the diagram or the question stem showed that O was actually at the origin (0,0). How can we prove that O is really at (0,0)?

Thank

two things here --

(1) when the letter "O" is placed at the center of a coordinate system, you can always trust that this point represents the origin. in fact, this assumption is stated on page number 135 of the official guide ("The point at which these two axes intersect, designated O, is called the origin.")

(2) even if you didn't have that assumption in this problem, you would still know that O stands for the origin -- because there's nothing else that it can stand for!
i.e., there are no other intersections or marked points in the neighborhood, so there are no competing definitions for where O is located.