Tim is a mathematician with two rather intelligent children, Matthew and Kristen. One day, he offers them the following challenge.
He tells them that he has a triangle whose side lengths are all integers. He tells only Matthew that the perimeter is 11 and tells only Kristen the area. Each is asked to independently determine the three side lengths of the triangle and each is intelligent enough to do so if they are given sufficient information.
After a few minutes Tim asks each of the children if they have come up with the answer and each indicates that it is impossible to determine with certainty. At that point, Matthew thinks for a moment and with no further information says that he now knows the answer with certainty. At that point, Kristen then thinks for a moment and indicates that she now knows with certainty.
What were the lengths of the three sides of the triangle?
*You must show sufficient work to support your answer.
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