# Midpoint Theorem

Theorem: The line segment joining the midpoints of two sides of a triangle is parallel to the third side and equal to half of it.

Given: A triangle PQR in which S and T are the midpoint of PQ and PR respectively.

To prove: ST ∥ QR and ST = $$\frac{1}{2}$$QR

Construction: Draw RU ∥ QP such that RU meets ST produced at U. Join SR.

Proof:

 Statement Reason 1. In ∆PST and ∆RUT,(i) PT = TR(ii) ∠PTS = ∠RTU(iii) ∠SPT = ∠TRU 1.(i) T is the midpoint of PR.(ii) Vertically opposite angles.(iii) Alternate angles. 2. Therefore, ∆PST ≅ ∆RUT 2. By AAS criterion of congruency. 3. Therefore, PS = RU; ST = TU 3. CPCTC. 4. But PS = QS 4. S is the midpoint of PQ. 5. Therefore, RU = QS and QS ∥ RU. 5. From statements 3, 4 and construction. 6. In ∆SQR and ∆RUS, ∠QSR = ∠URS, QS = RU. 6. From statement 5. 7. SR = SR. 7. Common side 8.  ∆SQR ≅ ∆RUS. 8. SAS criterion of congruency. 9. QR = SU = 2ST and ∠QRS = ∠RSU 9. CPCTC and statement 3. 10. ST = $$\frac{1}{2}$$QR and ST ∥ QR 10. By statement 9.