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Question

PQRS is a parallelogram. L and M are points on PQ and SR respecitvely such that PL = MR. Show that LM and QS bisect each other. 


Solution

Let LM and BAD intersect at O

as AL = CM and AB = CD
therefore AB-AL=CD-CM
BL = DM ___ (i)

consider  ΔOMD and  ΔOLB
  DOM=BOL (vertically opposite angles)
OMD=OLB     (alternate interior angles)
 DM = BL (from i)
therefore     Δ OMD congruent to  ΔOLB
therefore OM = OL,

OD = OB (cpctc)

Therefore O bisects LM and BD
therefore LM, BD bisect each other

(here parellelogram is ABCD)

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