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 parallel to $overline{fg}$, line t intersects $overline{fg}$ at a right angle, line t is perpendicular to $overline{fg}$, $fg = fh$, point h is the mid - point of $overline{fg}$

Explanation:

Step1: Recall definition of perpendicular bisector

A perpendicular bisector of a line segment is a line that is perpendicular to the segment and divides it into two equal parts.

Step2: Analyze option A

Since line t is a perpendicular bisector of $\overline{FG}$, it cannot be parallel to $\overline{FG}$. Parallel lines do not intersect, while a perpendicular bisector intersects the segment. So option A is false.

Step3: Analyze option B

By the definition of a perpendicular bisector, it intersects the line segment at a right angle. So option B is true.

Step4: Analyze option C

"Perpendicular bisector" implies that the line is perpendicular to the segment. So line t is perpendicular to $\overline{FG}$, and option C is true.

Step5: Analyze option D

Since H is the point of intersection of the perpendicular bisector and $\overline{FG}$, and the perpendicular bisector divides $\overline{FG}$ into two equal parts, $FG = 2FH$, not $FG=FH$. So option D is false.

Step6: Analyze option E

A bisector divides a line segment into two equal parts. So the point of intersection H of the perpendicular bisector t and $\overline{FG}$ is the mid - point of $\overline{FG}$. Option E is true.

Answer:

• B. Line t intersects $\overline{FG}$ at a right angle
• C. Line t is perpendicular to $\overline{FG}$
• E. Point H is the mid - point of $\overline{FG}$