Question

line ( t ) is the perpendicular bisector of ( overline{fg} ). if line ( t ) intersects ( overline{fg} ) at point ( h ), which of the following statements must be true?
check all that apply.

• line ( t ) is perpendicular to ( overline{fg} )
• point ( h ) is the midpoint of ( overline{fg} )
• line ( t ) is parallel to ( overline{fg} )
• line ( t ) intersects ( overline{fg} ) at a right angle
• ( fg = fh )

Explanation:

Step1: Recall Perpendicular Bisector Definition

A perpendicular bisector of a segment is a line that is perpendicular to the segment and bisects it (passes through its midpoint). So, by definition, line \( t \) (the perpendicular bisector of \( \overline{FG} \)) must be perpendicular to \( \overline{FG} \) and pass through its midpoint (so \( H \) is the midpoint of \( \overline{FG} \)) and intersect \( \overline{FG} \) at a right angle.

Step2: Analyze Each Statement

• Line \( t \) is perpendicular to \( \overline{FG} \): By definition of perpendicular bisector, this is true.
• Point \( H \) is the midpoint of \( \overline{FG} \): A bisector (perpendicular bisector) passes through the midpoint, so since \( t \) bisects \( \overline{FG} \) at \( H \), \( H \) is the midpoint. True.
• Line \( t \) is parallel to \( \overline{FG} \): A perpendicular bisector is perpendicular, not parallel, to the segment. False.
• Line \( t \) intersects \( \overline{FG} \) at a right angle: Perpendicular lines intersect at right angles, and \( t \) is perpendicular to \( \overline{FG} \), so this is true.
• \( FG = FH \): \( FH \) is half of \( FG \) (since \( H \) is the midpoint), so \( FG = 2FH \), not \( FG = FH \). False.

Answer:

• Line \( t \) is perpendicular to \( \overline{FG} \)
• Point \( H \) is the midpoint of \( \overline{FG} \)
• Line \( t \) intersects \( \overline{FG} \) at a right angle