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In a Δ ABC, P and Q are respectively the mid-points of AB and BC and R is the mid-point of AP. Prove that:
(i) ar (Δ PBQ) = ar (Δ ARC)
(ii) ar (Δ PRQ) = 1212 ar (Δ ARC)
(iii) ar (Δ RQC) = 3838 ar (Δ ABC).



Answer -

Given:
(1) In a triangle ABC, P is the mid-point of AB.
(2) Q is mid-point of BC.
(3) R is mid-point of AP.
To prove:
(a) Area of ΔPBQ = Area of ΔARC
(b) Area of ΔPRQ =  Area of ΔARC
(c) Area of ΔRQC =   Area of ΔABC
Proof: We know that each median of a triangle divides it into two triangles of equal area.
(a) Since CR is a median of ΔCAP
Therefore   …… (1)
Also, CP is a median of ΔCAB.
Therefore   …… (2)
From equation (1) and (2), we get
Therefore   …… (3)
PQ is a median of ΔABQ
Therefore  
Since  
Put this value in the above equation we get
 
  …… (4)
From equation (3) and (4), we get
Therefore  …… (5)
(b)  
  …… (6)
  …… (7)
From equation (6) and (7)
  …… (8)
From equation (7) and (8)
 
(c)  
 
=   …… (9)




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