Proving a Midpoint
Part of the Congruence & Proof section of Coddy's Geometry journey. Lesson 35 of 63.
A point is the midpoint of a segment when it lies on the segment and cuts it into two equal halves. As with the bisector, the Definition of midpoint works in both directions. If M is the midpoint of AB, then AM = MB. If M is on AB and AM = MB, then M is the midpoint.
| Statement | Reason |
|---|---|
| 1. AM = MB | Given (the ticks) |
| 2. M is the midpoint of AB | Definition of midpoint, using line 1 |
In a proof the halves are rarely marked like this. They are usually matching sides of two triangles, so the plan is the one from the bisector lesson: prove the triangles congruent, take the halves by CPCTC, then cite the definition. On the board, a midpoint statement is written by tapping the two halves of the segment.
Challenge
In △PQR the ticks show QP = QR, and QS meets PR at S. The two angles at Q are marked equal. Prove that S is the midpoint of PR, 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 CPCTC2Congruent 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