Question

Complete the missing parts of the paragraph proof.

Draw a perpendicular from P to AB. Label the intersection C. We are given that PA = PB, so PA ≅ PB by the definition of
. We know that angles PCA and PCB are right angles by the definition of
. PC ≅ PC by the
. So, triangle ACP is congruent to triangle BCP by HL, and AC ≅ BC by
. Since PC is perpendicular to and bisects AB, P is on the perpendicular bisector of AB by the definition of perpendicular bisector.

Given: P is equidistant from the endpoints of AB.

Prove: P lies on the perpendicular bisector of AB.

Point P is above line A B. A right arrow goes from this figure to triangle A P B. A dotted line is drawn from point P to the midpoint of side A B at point C.

Answer 1

Draw a perpendicular from P to AB. Label the intersection C. We are given that PA = PB, so PA ≅ PB by the definition of congruent segments. We know that angles PCA and PCB are right angles by the definition of perpendicular lines. PC ≅ PC by the reflexive property. So, triangle ACP is congruent to triangle BCP by HL, and AC ≅ BC by CPCTC (corresponding parts of congruent triangles are congruent). Since PC is perpendicular to and bisects AB, P is on the perpendicular bisector of AB by the definition of perpendicular bisector.