<p>Points P, Q, and R lie in a plane. If the distance between P and Q is 5 and the distance between Q and R is 2, which of the following could be the distance between P and R?</p>
<p>I. 3
II. 5
III. 7</p>
<p>A) I only
B) II only
C) III only
D) I and III only
E) I, II, and III</p>
<p>I don't know how to go about solving this problem. Can someone help me? The correct answer is E.</p>
<p>…I would have answered D</p>
<p>This is a “triangle rule” problem. The triangle rule says that the third side of a triangle is strictly between the difference and sum of the other 2 sides.</p>
<p>In this case, the difference is 5-2=3, and the sum is 5+2=7.</p>
<p>So the third side of a triangle x would have to satisfy 3<x<7. </p>
<p>But in this problem a straight line is allowed as well, so x could be 3 or 7 as well.</p>
<p>So the answer is E.</p>
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<p>I) Is possible. This would result in a straight line. Point R would lie on line PQ, and PR + QR = PQ (2 + 3 = 5)
II) Is possible. This would result in an isosceles triangle. Its points all lie in the same plane and all the sides are connected.
III) Is possible. This would also result in a straight line. Point Q would lie on line PR, and PQ + QR = PR (2 + 5 = 7)</p>
<p>I don’t understand why the distance between P and R can be either 3 or 7! Shouldn’t the distance be a number between 3 and 7 ?</p>
<p>Draw a segment of length 5 connected to a segment of length 2 to form a segment of length 7.</p>
<p>Draw a segment of length 5, and then chop off a segment of length 2 to form a segment of length 3.</p>
<p>The numbers strictly between 3 and 7 create triangles. The extreme values create straight lines.</p>