Given:& PA=PB with PM ⊥ AB at M.
Prove:& P is on the perpendicular bisector
&of AB.
We will begin by stating the given statement.
Given
PA=PB with PM ⊥ AB at M.
Recall that if two line segments have the same length, then they are congruent.
Definition of Segment Congruence
PA≅PB
Since PM ⊥ AB at M, we can say that ∠ PMA and ∠ PMB are right angles by the definition of perpendicular lines.
Definition of Perpendicular Lines
m∠ PMA=m∠ PMB=90^(∘)
Recall that all right angles are congruent.
Definition of Angle Congruence
∠ PMA≅∠ PMB
By the definition of right triangle, we can say that △ PMA and △ PMB are right triangles.
Definition of Right Triangle
△ PMA and △ PMB are right triangles.
Considering the Reflexive Property of Congruence, PM is congruent to itself.
Reflexive Property of Congruence
PM≅PM
Since the hypotenuse and one of the legs of △ PMA are congruent to the hypotenuse and one of the legs of △ PMB, these two triangles are congruent by the Hypotenuse-Leg Theorem.
Hypotenuse-Leg Theorem
△ PMA≅△ PMB
Knowing that the corresponding parts of congruent triangles are congruent, we can say that AM is congruent to BM.
Definition of Congruent Triangles
AM ≅ BM
Recall that if the line segments are congruent, then they have the same length.
Definition of Segment Congruence
AM=BM
Notice that the distance from A to M is the same as the distance from M to B. Therefore, we can deduce that M is the midpoint of AB.
Definition of Midpoint
M is the midpoint of AB
So far, we have proved that PM perpendicularly bisects AB. Therefore, PM is the perpendicular bisector of AB.
Definition of Perpendicular Bisector
PM is the perpendicular bisector of AB.
Since P is on PM, we can complete our proof.
Point on a Line
P is on the perpendicular bisector of AB.
Combining all of these steps, we can form our two-column proof.
