Certainly, let's analyze each statement one by one to determine which must be true given that line [tex]\( s \)[/tex] is the perpendicular bisector of line segment [tex]\( \overline{JK} \)[/tex] and intersects [tex]\( \overline{JK} \)[/tex] at point [tex]\( L \)[/tex].
Statement A: Point [tex]\( L \)[/tex] is the midpoint of [tex]\( \overline{JK} \)[/tex]
Since line [tex]\( s \)[/tex] is the perpendicular bisector of [tex]\( \overline{JK} \)[/tex], it divides [tex]\( \overline{JK} \)[/tex] into two equal parts. This means point [tex]\( L \)[/tex] must be the midpoint of segment [tex]\( \overline{JK} \)[/tex].
Thus, Statement A is True.
Statement B: Line [tex]\( s \)[/tex] intersects [tex]\( \overline{JK} \)[/tex] at a [tex]\(180^\circ\)[/tex] angle
A perpendicular bisector intersects the segment it bisects at a right angle ([tex]\(90^\circ\)[/tex]), not a straight line angle ([tex]\(180^\circ\)[/tex]).
Thus, Statement B is False.
Statement C: Line [tex]\( s \)[/tex] is parallel to [tex]\( \overline{JK} \)[/tex]
A perpendicular bisector cannot be parallel to the segment it bisects because it intersects the segment at a right angle.
Thus, Statement C is False.
Statement D: Line [tex]\( s \)[/tex] is perpendicular to [tex]\( \overline{JK} \)[/tex]
By definition, a perpendicular bisector intersects the segment it bisects at a right angle ([tex]\(90^\circ\)[/tex]).
Thus, Statement D is True.
Statement E: [tex]\( JL = KL \)[/tex]
Since point [tex]\( L \)[/tex] is the midpoint of [tex]\( \overline{JK} \)[/tex], the lengths of [tex]\( \overline{JL} \)[/tex] and [tex]\( \overline{KL} \)[/tex] are equal.
Thus, Statement E is True.
Summarizing, the true statements are:
- A. Point [tex]\( L \)[/tex] is the midpoint of [tex]\( \overline{JK} \)[/tex].
- D. Line [tex]\( s \)[/tex] is perpendicular to [tex]\( \overline{JK} \)[/tex].
- E. [tex]\( JL = KL \)[/tex].
The final applicable answers are:
True, False, False, True, True.
