Official Guide Explanation:
Problem Solving #150




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Solution and Metadata

Question: 150
Page: 82
Difficulty: 6 (Moderately Difficult)
Category 1: Geometry > Polygons (Convex) >

Explanation:     Because the diagram is not drawn to scale, we don't know anything about the angles. If QPT and PTS are both very large angles, the length of PT could be very short. But if QPT and PTS are very small, acute angles, PT could be very long. We can use a version of the rules you should know regarding the length of the third side of a triangle. The third side of a triangle cannot be longer than the sum of the other two sides.

The same is true here--PT cannot be longer than the sum of the other four sides. If you tried to draw a figure with a length of PT like that, it would be impossible to connect the other four sides together. The sum of the other four sides is 14, so PT cannot be 15. The other side lengths are acceptable. As we've seen, PT can be very small relative to the other side lengths, so 5 would be easy to construct. 10 is under the limit of 14, so 5 and 10 are both possible. Choice (C) is correct.

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