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.


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

Midpoint Theorem on Right-angled Triangle

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

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




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


(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°.


(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.


8. Therefore, QS = \(\frac{1}{2}\)PR.

8. Using statement 7 in statement 1.

9th Grade Math

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