Question

ABCD is a qudrilateral in wch P,Q,R and S are mid-points of the sides AB,BC,CD and DA. AC is a diagonal show that (i)SR||AC and SR = (1/2)AC
(ii) PQ = SR
(iii) PQRS is a parallelogram.

Answer 1

hii 
here is your answer

We have parallelogram ABCD. BD is a diagonal and ‘P’ and ‘Q’ are such that                PD = QB[Given]           (i) To prove that ΔAPD ≌ ΔCQB                ∵ AD || BC and BD is a transversal.[∵ABCD is a parallelogram.]                ∴∠ADB = ∠CBD[Interior alternate angles]                ⇒∠ADP = ∠CBQ               
 Now, in ΔAPD and ΔCQB, we have               
 AD =CB[Opposite side of the parallelogram]               
 PD = QB[Given]              
  ∠CBQ = ∠ADP[Proved]              
  ∴ Using SAS criteria, we have              
  ΔAPD ≌ ΔCQB          
 (ii) To prove that AP = CQ                
Since ΔAPD ≌ ΔCQB[Proved]            
    ∴ Their corresponding parts are equal.           
     ⇒AP = CQ           (iii) To prove that ΔAQB ≌ ΔCPD.        
        AB || CD and BD is a transversal.[ ∵ ABCD is a parallelogram.]   
             ∴∠ABD = ∠CDB              
  ⇒∠ABQ = ∠CDP              
  Now, in ΔAQB and ΔCPD, we have QB = PD[Given] ∠ABQ = ∠CDP[Proved]
 AB = CD[Opposite sides of parallelogram ABCD] 
∴ΔAQB ≌ ΔCPD[SAS criteria]         
  (iv) To prove that AQ = CP.        
        Since              
  ΔAQB ≌ ΔCPD[Proved]          
      ∴Their corresponding parts are equal. 
               ⇒                AQ = CP.         
  (v) To prove that APCQ is a parallelogram. 
               Let us join AC.              
  Since, the diagonals of a || gm bisect each other   
             ∴AO = CO          
      and                           
     BO = DO               
 ⇒(BO – BQ) = (DO – DP)[∵ BQ = DP (Given)]       
         ⇒QO = PO...(2)           
     Now, in quadrilateral APCQ, we have          
                      AO = CO and QO = PO  
              i.e. AC and QP bisect each other at O.  
              ⇒APCQ is a parallelogram.