Question

plz solve this problem hurrily​

Answer 1

Since PS is the median, $QS=SR=(\frac{QR}{2})$.

(i) We can write

$TR = ST + SR$

Now,

$TR^2 = (ST +SR)^2 \\ =ST^2 + (2*SR)*ST + SR^2\\ = ST^2 + QR*ST + SR^2$..........(1)

Using Pythagoras theorem in ΔPST,

$PS^2=PT^2+ST^2$..........(2)

Using Pythagoras theorem in ΔPRT,

$PR^2= PT^2 + TR^2\\or, PR^2 =(PT^2+ ST^2)+QR*ST +SR^2[using (1)]\\ or, PR^2=PS^2+QR*ST +ST^2[using (2)]\\ or, PR^2=PS^2+ QR*ST+ (\frac{QR}{2})^2$

Hence, proved.

(ii)We can write

$QT=QS-ST$

Now,

$QT^2=(QS-ST)^2\\ =QS^2 -2*QS*ST +ST^2\\ =QS^2-QR*ST +ST^2$..........(3)

Using Pythagoras theorem in ΔPQT,

$PQ^2= PT^2+QT^2\\or, PQ^2 = PS^2-ST^2+QT^2 [using (2)]\\ or, PQ^2=PS^2-ST^2+QS^2-QR*ST+ST^2[using (3)]\\ or, PQ^2=PS^2-QR*ST+(\frac{QR}{2})^2$

Hence, proved.