Question 7 (iii) P and Q are respectively the mid-points of sides AB and BC of a triangle ABC and R is the mid-point of AP, Show that: ar(ΔPBQ)=ar(ΔARC).
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Construction: Join PQ, RQ and RC
CR is a median of ΔCAP. ⇒ar(ΔARC)=12ar(ΔCAP) ---- (1) Also, CP is a median of ΔABC. ⇒ar(ΔCAP)=ar(ΔCBP) ------ (2) From (1) and (2), we get, ⇒ar(ΔARC)=12ar(ΔCBP) ----- (3) PQ is a median of ΔPBC. ⇒ar(ΔCPB)=2ar(ΔPBQ) ----- (4) From (3) and (4), we get, ⇒ar(ΔARC)=ar(ΔPBQ) ----- (5) Hence, ar(ΔARC)=ar(ΔPBQ)