The Perpendicular Bisector
Part of the Congruence & Proof section of Coddy's Geometry journey. Lesson 50 of 63.
The perpendicular bisector of a segment is the line through its midpoint at right angles to it. Every point on it is the same distance from both ends of the segment. Proving that uses two definitions at once: the Definition of midpoint gives equal halves, and the Definition of perpendicular turns perpendicular lines into right angles.
| Statement | Reason |
|---|---|
| 1. CD ⊥ AB | Given |
| 2. ∠CDA is a right angle | Definition of perpendicular, using line 1 |
| 3. ∠CDB is a right angle | Definition of perpendicular, using line 1 |
Here CD ⊥ AB, and the definition turns that one fact into two right angles at D, one line for each. To prove the theorem you will combine those right angles with the midpoint's halves and the side the two triangles share. The proof is long, but each step is a reason you already know.
Challenge
You are given that S is the midpoint of PQ and that RS ⊥ PQ. Prove that RP = RQ, then press Check.
Try it yourself
Write the proof
Test yourselfThis lesson includes a short quiz. Start the lesson to answer it and track your progress.
All lessons in Congruence & Proof
1What a Proof Is
Given and ProveStatements and ReasonsChaining Equal PartsYour First Full ProofRecap: A Chain of Reasons7Using CPCTC
Congruent, Then EqualProving a BisectorFlowchart ProofsProving a MidpointRecap: Using CPCTC10Isosceles Triangle Proofs
Base Angles in a ProofEqual Angles, Equal SidesProving IsoscelesThe Perpendicular BisectorChasing in Isosceles ShapesRecap: Isosceles Proofs2Congruent Triangles
Same Size, Same ShapeMatching SidesMatching AnglesWhat Congruence GivesRecap: Matching Parts8Proofs with Vertical Angles
Vertical Angles in ProofsThe BowtieBowtie with ASALinear Pairs in ProofsRecap: Vertical Angles