In triangle $ABC$ , $AD$ is perpendicular to $BC$. It is given that $BC$=8 cm and $AD$=6 cm. If $E$ and $F$ are the mid points of $BD$ and $AC$ respectively, find $EF$.
I tried using the mid point theorem by constructing a line parallel to $AB$, but the equations led nowhere.
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$\begingroup$Let $M$ denote the midpoint of the segment $AB$. Then the midpoint theorem implies that $ME$ has length 3 (by comparison with the length of the segment $AD$), and another application of the midpoint theorem implies that $MF$ has length 4 (by comparison with the length of the segment $BC$). Because $ME$ is parallel to $AD$, $MF$ is parallel to $BC$, and $AD$ is perpendicular to $BC$, we have that $ME$ and $MF$ are perpendicular. Thus, $MEF$ is a right triangle with legs of length 3 and 4 so that the hypotenuse $EF$ has length 5.
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