In the figure 2.28 seg PS is the median of PQR and PT⊥QR. Prove that,

(1) QS = ½ QR ……(i) [S is the midpoint of QR]
SR = ½ QR ……(ii)
QS = SR [From (i) and (ii)]
PT ⊥QR [Given]
PSR is an obtuse angle. [From figure]
PR² = SR²+PS²+2SR×ST …..(iii) [Application of Pythagoras theorem]
Substitute SR = ½ QR in (iii)
PR² =[(½)QR]²+PS²+2(1/2)QR×ST
PR² =[(½)QR]²+PS²+QR×ST
PR² = PS² + QR×ST +(QR/ 2 )²
Hence proved.

(ii) PT⊥QS [Given]

PSQ is an acute angle [From figure]
PQ² = QS²+PS²-2QS×ST [Application of Pythagoras theorem]
PR² = [(1/2)QR]²+PS²-2(1/2 )QR×ST
PR² = (QR/2)²+PS²-QR×ST
PR² = PS²-QR×ST+( QR /2)²
Hence proved.