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.

Midpoint Theorem on Trapezium

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.






9th Grade Math

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