Question

prove that in a right angle triangle the length of the median of the hypotenuse is half of the length of the hypotenuse​

Answer 1

Answer:

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

Midpoint Theorem on Right-angled Triangle

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To prove: QS = 12PR.

Answer 2

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

Midpoint Theorem on Right-angled Triangle

To prove: QS = 12PR.

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

1. In ∆PQR, PS = 12PR. [S is the midpoint of PR].

2.In ∆PQR,

(i) S is the midpoint of PR  [Given]

(ii) ST ∥ QR  [By construction.]

3.Therefore, T is the midpoint of PQ.  [By converse of the Midpoint Theorem.]

4.TS ⊥ PQ.  [TS ∥ QR and QR ⊥ PQ]In ∆PTS and ∆QTS ,

 5. (i) PT = TQ   [From the statement 3.]

(ii) TS = TS [ Common side.]

(iii) ∠PTS = ∠QTS = 90°. [From the statement 4.]

6. Therefore, ∆PTS ≅ ∆QTS. [ By SAS criterion of congruency].

7. PS = QS. [ CPCTC]

Therefore, QS = 12PR. [ Using statement 7 in statement 1.]