Question

1. in the diagram below, line mn is a perpendicular bisector. which of the following is true? options: \\(\overline{mt} \cong \overline{ms}\\), \\(\overline{mn} \cong \overline{qn}\\), \\(\overline{ms} \cong \overline{mn}\\), \\(\overline{mt} \cong \overline{mn}\\)

Explanation:

Step1: Recall Perpendicular Bisector Property

A perpendicular bisector of a segment is a line that is perpendicular to the segment and divides it into two congruent parts. So, if \( MN \) is the perpendicular bisector of \( ST \), then \( N \) is the midpoint of \( ST \), meaning \( \overline{SN} \cong \overline{TN} \). Also, any point on the perpendicular bisector is equidistant from the endpoints of the segment it bisects. Here, point \( M \) and point \( Q \) are on the perpendicular bisector \( MN \) (wait, actually \( MN \) is the perpendicular bisector of \( ST \), so points \( M \) and \( Q \) are on the line \( MQ \) which is the perpendicular bisector? Wait, looking at the diagram, \( MN \) is a perpendicular bisector, so \( SN = TN \) (the red marks) and \( \angle MNS=\angle MNT = 90^\circ \). Now, for the segments: Let's check the options.

Option 1: \( \overline{MT} \cong \overline{MS} \). Since \( M \) is on the perpendicular bisector of \( ST \), by the Perpendicular Bisector Theorem, any point on the perpendicular bisector is equidistant from \( S \) and \( T \). Wait, no, \( ST \) is bisected by \( MN \), so \( N \) is midpoint of \( ST \), and \( MN \perp ST \). Then, triangles \( MNS \) and \( MNT \) are congruent (SAS: \( SN = TN \), \( \angle MNS=\angle MNT \), \( MN = MN \)), so \( MS = MT \). Wait, but let's check the options. Wait the options are:

1. \( \overline{MT} \cong \overline{MS} \)
2. \( \overline{MN} \cong \overline{QN} \) – no info about \( QN \)
3. \( \overline{MS} \cong \overline{MN} \) – no, \( MS \) is a side of the triangle, \( MN \) is a segment, lengths not necessarily equal
4. \( \overline{MT} \cong \overline{MN} \) – same as above, not necessarily equal.

Wait, the Perpendicular Bisector Theorem: If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. Here, \( M \) is on the perpendicular bisector of \( ST \) (since \( MN \) is the perpendicular bisector of \( ST \), so line \( MQ \) (where \( Q \) is the bottom point) is the perpendicular bisector). So \( M \) is equidistant from \( S \) and \( T \), so \( MS = MT \), which is option 1: \( \overline{MT} \cong \overline{MS} \).

Wait, but let's re-examine the diagram. The diagram has \( S \) and \( T \) with \( SN = TN \) (red marks), \( MN \perp ST \), and \( MS = MT = 12 \) (blue marks). So \( MS \) and \( MT \) are both 12, so they are congruent. So the correct option is the first one: \( \overline{MT} \cong \overline{MS} \).

Step2: Verify Each Option

• Option 1: \( \overline{MT} \cong \overline{MS} \): Since \( M \) is on the perpendicular bisector of \( ST \), \( MS = MT \) (Perpendicular Bisector Theorem), so this is true.
• Option 2: \( \overline{MN} \cong \overline{QN} \): No information given about \( QN \), so can't conclude.
• Option 3: \( \overline{MS} \cong \overline{MN} \): \( MS \) is a slant side, \( MN \) is a vertical segment, no reason to be congruent.
• Option 4: \( \overline{MT} \cong \overline{MN} \): Same as above, no reason to be congruent.

Answer:

\(\overline{MT} \cong \overline{MS}\) (the first option)