# Midpoint Theorem on Trapezium

PQRS is a trapezium in which PQ ∥ RS. T is the midpoint of QR. TU is drawn parallel to PQ which meets PS  at U. Prove that 2TU = PQ + RS.

Given: PQRS is a trapezium in which PQ ∥ RS. T is the midpoint of QR. TU ∥ PQ and TU meets PS at U.

To prove: 2TU = PQ + RS.

Construction: Join QS. QS and TU intersect at M.

Proof:

 Statement Reason 1. PQ ∥ RS and TU ∥ PQ. 1. Given. 2. RS ∥ TU. 2. From statement 1. 3. In ∆QRS,T is the midpoint of QR and TM ∥ RS⟹ M is the midpoint of QS. 3. By the converse of the Midpoint Theorem. 4. In ∆PSQ,M is the midpoint of QS and MU ∥ PQ.⟹ U is the midpoint of PS. 4. By the converse of the Midpoint Theorem. 5. In ∆QRS, the line segment TM joining the midpoints of sides QR and QS.Therefore, TM = $$\frac{1}{2}$$RS. 5. By the Midpoint Theorem. 6. In ∆PQS, the line segment MU joins the midpoints of the sides QS and PS.Therefore, MU = $$\frac{1}{2}$$PQ. 6. By the Midpoint Theorem. 7. TM + MU = $$\frac{1}{2}$$RS + $$\frac{1}{2}$$PQ. 7. From statements 5 and 6. 8. TU = $$\frac{1}{2}$$(RS + PQ). 8.  TM + MU = TU. 9. 2TU = RS + PQ. (Proved) 9. From statement 8.