From the information given in the figure 2.31, prove that PM = PN = √3×a

Proof:

In PRM
Given MQ = QR = a
Q is the midpoint of MR .
PQ is the median.
PR²+PM² = 2PQ²+2QM² [Apollonius theorem]
a²+PM² = 2a²+2a²
PM² = 3a²
PM = √3a…………(i)

In PQN
Given NR = QR = a
R is the midpoint of QN.
PR is the median.
PN²+PQ² = 2PR²+2RN² [Apollonius theorem]
PN²+a² = 2a²+2a²
PN² = 3a²
PN = √3a…………..(ii)
From (i) and (ii) PM = PN = √3×a
Hence proved.