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:
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Statement |
Reason |
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1. PQ ∥ RS and TU ∥ PQ. |
1. Given. |
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2. RS ∥ TU. |
2. From statement 1. |
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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. |
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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. |
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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. |
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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. |
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7. TM + MU = \(\frac{1}{2}\)RS + \(\frac{1}{2}\)PQ. |
7. From statements 5 and 6. |
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8. TU = \(\frac{1}{2}\)(RS + PQ). |
8. TM + MU = TU. |
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9. 2TU = RS + PQ. (Proved) |
9. From statement 8. |
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