Answer :
Let's analyze each statement given that line [tex]\( t \)[/tex] is the perpendicular bisector of [tex]\( \overline{FG} \)[/tex] and it intersects [tex]\( FG \)[/tex] at point [tex]\( H \)[/tex].
Statement A: [tex]\( FG = FH \)[/tex]
For line [tex]\( t \)[/tex] to be the perpendicular bisector of [tex]\( FG \)[/tex], it means that it cuts [tex]\( FG \)[/tex] into two equal segments at [tex]\( H \)[/tex]. However, [tex]\( FG \)[/tex] is the entire segment, and [tex]\( FH \)[/tex] is only half of it. Therefore, [tex]\( FG \)[/tex] will not equal [tex]\( FH \)[/tex] unless [tex]\( H \)[/tex] coincides with [tex]\( G \)[/tex]. This does not hold true in general cases, so this statement is False.
Statement B: Point [tex]\( H \)[/tex] is the midpoint of [tex]\( \overline{FG} \)[/tex]
A perpendicular bisector of a segment always intersects the segment at its midpoint. Thus, [tex]\( H \)[/tex] must be the midpoint of [tex]\( \overline{FG} \)[/tex]. Therefore, this statement is True.
Statement C: Line [tex]\( t \)[/tex] is parallel to [tex]\( \overline{FG} \)[/tex]
A line that is a perpendicular bisector of a segment cannot be parallel to the segment it bisects because "perpendicular" implies it intersects the segment at a right angle. Thus, this statement is False.
Statement D: Line [tex]\( t \)[/tex] intersects [tex]\( \overline{FG} \)[/tex] at a right angle
By definition, a line that is a perpendicular bisector intersects the segment it bisects at a right angle (90 degrees). Hence, this statement is True.
Statement E: Line [tex]\( t \)[/tex] is perpendicular to [tex]\( \overline{FG} \)[/tex]
Since [tex]\( t \)[/tex] is described as the perpendicular bisector, it must necessarily be perpendicular to [tex]\( FG \)[/tex] by definition. Therefore, this statement is True.
In conclusion, the statements that must be true are:
- B. Point [tex]\( H \)[/tex] is the midpoint of [tex]\( \overline{FG} \)[/tex]
- D. Line [tex]\( t \)[/tex] intersects [tex]\( \overline{FG} \)[/tex] at a right angle
- E. Line [tex]\( t \)[/tex] is perpendicular to [tex]\( \overline{FG} \)[/tex]
Hence, the resulting truth values of the statements are:
- A: False
- B: True
- C: False
- D: True
- E: True
Statement A: [tex]\( FG = FH \)[/tex]
For line [tex]\( t \)[/tex] to be the perpendicular bisector of [tex]\( FG \)[/tex], it means that it cuts [tex]\( FG \)[/tex] into two equal segments at [tex]\( H \)[/tex]. However, [tex]\( FG \)[/tex] is the entire segment, and [tex]\( FH \)[/tex] is only half of it. Therefore, [tex]\( FG \)[/tex] will not equal [tex]\( FH \)[/tex] unless [tex]\( H \)[/tex] coincides with [tex]\( G \)[/tex]. This does not hold true in general cases, so this statement is False.
Statement B: Point [tex]\( H \)[/tex] is the midpoint of [tex]\( \overline{FG} \)[/tex]
A perpendicular bisector of a segment always intersects the segment at its midpoint. Thus, [tex]\( H \)[/tex] must be the midpoint of [tex]\( \overline{FG} \)[/tex]. Therefore, this statement is True.
Statement C: Line [tex]\( t \)[/tex] is parallel to [tex]\( \overline{FG} \)[/tex]
A line that is a perpendicular bisector of a segment cannot be parallel to the segment it bisects because "perpendicular" implies it intersects the segment at a right angle. Thus, this statement is False.
Statement D: Line [tex]\( t \)[/tex] intersects [tex]\( \overline{FG} \)[/tex] at a right angle
By definition, a line that is a perpendicular bisector intersects the segment it bisects at a right angle (90 degrees). Hence, this statement is True.
Statement E: Line [tex]\( t \)[/tex] is perpendicular to [tex]\( \overline{FG} \)[/tex]
Since [tex]\( t \)[/tex] is described as the perpendicular bisector, it must necessarily be perpendicular to [tex]\( FG \)[/tex] by definition. Therefore, this statement is True.
In conclusion, the statements that must be true are:
- B. Point [tex]\( H \)[/tex] is the midpoint of [tex]\( \overline{FG} \)[/tex]
- D. Line [tex]\( t \)[/tex] intersects [tex]\( \overline{FG} \)[/tex] at a right angle
- E. Line [tex]\( t \)[/tex] is perpendicular to [tex]\( \overline{FG} \)[/tex]
Hence, the resulting truth values of the statements are:
- A: False
- B: True
- C: False
- D: True
- E: True
