# Midpoint Theorem on Right-angled Triangle

Here we will prove that in a right-angled triangle the median drawn to the hypotenuse is half the hypotenuse in length.

Solution:

Given: In ∆PQR, ∠Q = 90°. QD is the median drawn to hypotenuse PR.

To prove: QS = $$\frac{1}{2}$$PR.

Construction: Draw ST ∥ QR such that ST cuts PQ at T.

Proof:

 Statement Reason 1. In ∆PQR, PS = $$\frac{1}{2}$$PR. 1. S is the midpoint of PR. 2. In ∆PQR,(i) S is the midpoint of PR(ii) ST ∥ QR 2.(i) Given.(ii) By construction. 3. Therefore, T is the midpoint of PQ. 3. By converse of the Midpoint Theorem. 4. TS ⊥ PQ. 4. TS ∥ QR and QR ⊥ PQ 5. In ∆PTS and ∆QTS ,(i) PT = TQ(ii) TS = TS(iii) ∠PTS = ∠QTS = 90°. 5.(i) From the statement 3.(ii) Common side.(iii) From the statement 4. 6. Therefore, ∆PTS ≅ ∆QTS. 6. By SAS criterion of congruency. 7. PS = QS. 7. CPCTC 8. Therefore, QS = $$\frac{1}{2}$$PR. 8. Using statement 7 in statement 1.

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