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I come across this question in a GRE practice book. The correct answer is A. But my first instinct was D. Because in the question, no information regarding the length of the sides are given. So the sign of the expression QR-PQ is indeterminate.

I read the GRE test guide, but I did not find any information addressing the condition that the sign can be ignored.

gre

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    $\begingroup$ Where is the actual question? $\endgroup$ – Arthur Oct 13 '13 at 9:00
  • $\begingroup$ the figure is the question. if you have seen GRE test questions before, you should know what A, B, C, D means... $\endgroup$ – David S. Oct 14 '13 at 8:05
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Ok, so after searching I have found that the possible answers are the following:

A) $PR > QR - PQ$
B) $PR < QR - PQ$
C) $PR = QR - PQ$
D) None of the above are in general true.

Answer A is equivalent with the statement $PR + PQ > QR$. This is known as the triangle inequality, and holds for an arbitrary triangle $PQR$. One way to see it is the following: $QR$ is the distance from $Q$ to $R$ in a straight line. This is obviously the shortest route from $Q$ to $R$, and so $QR$ is shorter than $PR + PQ$, which is the distance from $Q$ to $R$ via $P$.

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  • $\begingroup$ the answer would be much easier to see if it is given in the way you provided. but the problem here is, in the way it is given in the GRE test, I am not sure if I need to consider the sign of the option QR-PQ $\endgroup$ – David S. Oct 14 '13 at 13:17
  • $\begingroup$ The value $QR-PQ$ could well be negative, in that case the positive length $PR$ is surely larger than it. $\endgroup$ – Arthur Oct 14 '13 at 18:05

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