In triangle RTS with a right angle at R and MN as the perpendicular bisector of TS, it is proven that TM is a median by showing it bisects RS and passes through its midpoint.
To prove that \(TM\) is a median, we need to show that \(TM\) bisects the opposite side \(RS\) and that \(TM\) passes through the midpoint of \(RS\).
Given:
1. \(m\angle RTS = 90^\circ\)
2. \(MN\) is the perpendicular bisector of \(TS\)
To prove \(TM\) is a median:
1. Show \(TM\) bisects \(RS\):
Since \(MN\) is the perpendicular bisector of \(TS\), it implies that \(RMT\) and \(SMT\) are congruent right-angled triangles. Therefore, \(TM\) bisects \(RS\).
2. Show \(TM\) passes through the midpoint of \(RS\):
Since \(TM\) bisects \(RS\), it automatically passes through the midpoint of \(RS\)
Therefore, with the given conditions, \(TM\) is a median in triangle \(RTS\).