Question

question 23 of 25
line ( k ) is the perpendicular bisector of ( overline{pq} ). if line ( k ) intersects ( overline{pq} ) at point ( r ), which of the following statements must be true?
check all that apply.
a. line ( k ) bisects ( overline{pq} )
b. line ( k ) intersects ( overline{pq} ) at a ( 90^{circ} ) angle
c. ( overline{pr} cong overline{qr} )
d. point ( r ) is the midpoint of line ( k )
e. line ( k ) is parallel to ( overline{pq} )

Explanation:

• For option A: By the definition of a perpendicular bisector, a perpendicular bisector of a line segment bisects the segment. So line \(k\) bisects \(\overline{PQ}\).
• For option B: Since it is a perpendicular bisector, line \(k\) intersects \(\overline{PQ}\) at a \(90^{\circ}\) angle.
• For option C: If a line is the perpendicular bisector of a segment \(\overline{PQ}\) and intersects it at \(R\), then \(R\) is the mid - point of \(\overline{PQ}\), so \(\overline{PR}\cong\overline{QR}\) (by the definition of a mid - point, segments on either side of the mid - point are congruent).
• For option D: There is no information given to suggest that \(R\) is the mid - point of line \(k\). The perpendicular bisector \(k\) can be of any length, and \(R\) is just the point of intersection with \(\overline{PQ}\).
• For option E: A perpendicular bisector is perpendicular (not parallel) to the segment it bisects. So line \(k\) is not parallel to \(\overline{PQ}\).

Answer:

A. Line \(k\) bisects \(\overline{PQ}\), B. Line \(k\) intersects \(\overline{PQ}\) at a \(90^{\circ}\) angle, C. \(\overline{PR}\cong\overline{QR}\)