Question

In ΔPQR, LM = MN, QM = MR and ML and MN are perpendiculars on PQ and PR respectively. Prove that PQ = PR.

Answer 1

In ΔQLM and ΔRNM
QM = MR
LM = MN
∠QLM = ∠RNM = 90°
Therefore, ΔQLM ≅ ΔRNM  ...(RHS criteria)
Hence, QL = RN ..........(i)
Join PM
In ΔPLM and ΔPNM and 
PM = PM    ...(common)
LM = MN
∠PLM = ∠PNM = 90°
Therefore, ΔPLM ≅ ΔPNM   ...(RHS criteria)
Hence, PL = PN ..........(ii)
From (i) and (ii)
PQ = PR.