Since JL is the perpendicular bisector of KM, it means that the lengths of segments KH and HM are equal. Therefore, we can set the expressions for KH and HM equal to each other:
\[ 6y + 3 = 2y + 7 \]
Next, we will solve for \(y\).
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Subtract \(2y\) from both sides: \[ 6y - 2y + 3 = 7 \] \[ 4y + 3 = 7 \]
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Now, subtract 3 from both sides: \[ 4y = 7 - 3 \] \[ 4y = 4 \]
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Divide both sides by 4 to find \(y\): \[ y = 1 \]
Now that we have \(y\), we can substitute it back into the expression for \(HM\) (or \(KH\), since they are equal) to find the length of segment \(HM\):
Using the expression \(HM = 2y + 7\): \[ HM = 2(1) + 7 = 2 + 7 = 9 \]
Thus, the length of segment \(HM\) is:
\[ \boxed{9} \]
This corresponds with option c.