Talk:2009 AMC 10B Problems/Problem 12

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In the solution, it says that the area of a triangle will be uniquely determined by the base length, since the height is determined. However, I was wondering how do we know that the height is determined. The problem never specifically says that the line containing points $E$ and $F$ was a set distance away from the line containing the points $A$, $B$, $C$, $D$. Can't we have the lines $100$ units apart, and have a triangle with a base of $1$, and a height of $50$.

Please let me know what I am missing. Thanks!