Answer :
To determine which statements must be true given that line [tex]\( t \)[/tex] is the perpendicular bisector of [tex]\( \overline{FG} \)[/tex] and intersects [tex]\( \overline{FG} \)[/tex] at point [tex]\( H \)[/tex], let's analyze the properties of a perpendicular bisector.
1. Line [tex]\( t \)[/tex] is perpendicular to [tex]\( \overline{FG} \)[/tex]:
A perpendicular bisector is defined as a line that divides another line segment into two equal parts at a right angle (90 degrees). Therefore, line [tex]\( t \)[/tex] must be perpendicular to [tex]\( \overline{FG} \)[/tex].
True
2. Line [tex]\( t \)[/tex] is parallel to [tex]\( \overline{FG} \)[/tex]:
Since line [tex]\( t \)[/tex] is described as a perpendicular bisector, by definition, it cannot be parallel to [tex]\( \overline{FG} \)[/tex] because it intersects [tex]\( \overline{FG} \)[/tex] at a 90-degree angle.
False
3. Line [tex]\( t \)[/tex] intersects [tex]\( \overline{FG} \)[/tex] at a right angle:
As previously mentioned, a perpendicular bisector intersects the segment it bisects at a 90-degree angle.
True
4. Point [tex]\( H \)[/tex] is the midpoint of [tex]\( \overline{FG} \)[/tex]:
The characteristic of a bisector is to divide the segment into two equal parts. Hence, point [tex]\( H \)[/tex] is exactly in the middle of [tex]\( \overline{FG} \)[/tex], making [tex]\( H \)[/tex] the midpoint.
True
5. [tex]\( FG = FH \)[/tex]:
The segment [tex]\(\overline{FG}\)[/tex] is twice the length of [tex]\(\overline{FH}\)[/tex], not equal to [tex]\(\overline{FH}\)[/tex]. So, [tex]\( FG \)[/tex] is not equal to [tex]\( FH \)[/tex]; rather, [tex]\( FH \)[/tex] is half of [tex]\( FG \)[/tex].
False
The statements that must be true are:
1. A. Line [tex]\( t \)[/tex] is perpendicular to [tex]\( \overline{FG} \)[/tex]
3. C. Line [tex]\( t \)[/tex] intersects [tex]\( \overline{FG} \)[/tex] at a right angle
4. D. Point [tex]\( H \)[/tex] is the midpoint of [tex]\( \overline{FG} \)[/tex]
Hence, the correct answers are:
[tex]\[ [1, 3, 4] \][/tex]
1. Line [tex]\( t \)[/tex] is perpendicular to [tex]\( \overline{FG} \)[/tex]:
A perpendicular bisector is defined as a line that divides another line segment into two equal parts at a right angle (90 degrees). Therefore, line [tex]\( t \)[/tex] must be perpendicular to [tex]\( \overline{FG} \)[/tex].
True
2. Line [tex]\( t \)[/tex] is parallel to [tex]\( \overline{FG} \)[/tex]:
Since line [tex]\( t \)[/tex] is described as a perpendicular bisector, by definition, it cannot be parallel to [tex]\( \overline{FG} \)[/tex] because it intersects [tex]\( \overline{FG} \)[/tex] at a 90-degree angle.
False
3. Line [tex]\( t \)[/tex] intersects [tex]\( \overline{FG} \)[/tex] at a right angle:
As previously mentioned, a perpendicular bisector intersects the segment it bisects at a 90-degree angle.
True
4. Point [tex]\( H \)[/tex] is the midpoint of [tex]\( \overline{FG} \)[/tex]:
The characteristic of a bisector is to divide the segment into two equal parts. Hence, point [tex]\( H \)[/tex] is exactly in the middle of [tex]\( \overline{FG} \)[/tex], making [tex]\( H \)[/tex] the midpoint.
True
5. [tex]\( FG = FH \)[/tex]:
The segment [tex]\(\overline{FG}\)[/tex] is twice the length of [tex]\(\overline{FH}\)[/tex], not equal to [tex]\(\overline{FH}\)[/tex]. So, [tex]\( FG \)[/tex] is not equal to [tex]\( FH \)[/tex]; rather, [tex]\( FH \)[/tex] is half of [tex]\( FG \)[/tex].
False
The statements that must be true are:
1. A. Line [tex]\( t \)[/tex] is perpendicular to [tex]\( \overline{FG} \)[/tex]
3. C. Line [tex]\( t \)[/tex] intersects [tex]\( \overline{FG} \)[/tex] at a right angle
4. D. Point [tex]\( H \)[/tex] is the midpoint of [tex]\( \overline{FG} \)[/tex]
Hence, the correct answers are:
[tex]\[ [1, 3, 4] \][/tex]
