7. P and Q are respectively the mid-points of sides AB and BC of a triangk ABC and R is the mid-point of AP, show that(i) ar(PRQJ = -iar(ARCJ (ii) ar(RQCJ =iar(ABCJ(iii) ar(PBQJ = ar(ARCJ

Construction: Join A A and Q; P and C. Proof: QP is median of triangle BQA. :. ar(BQP) = ar(PQA) ...(i) Similarly, ar(QPR) = ar(RQA) [QR is median of triangle PQA] ar(PQA) = 2 ar(QPR) [From above result] .. 2 ar(QPR) = ar(BQP) [From (i)) 1 =-ar(BPC) 2 1 =2ar(APC) = a.r(ARC) [PQ is median] lCP is median] [RC is median] 1 => ar(QPR) = 2 ar(ARC). (ii) ar = jar = iar(RBC) = i far(ABC) - ar(ARC)) = j {ar(ABC)- ar(ABC)} = ar(ABC). 1 (iii) ar(ARC> = 2ar(CAP> ...(ii) [CR is median] And ar(CAP> = ar(CPB> ...(iii) [CP is median] From equations (ii) and (iii), we have 1 ar(ARC) = 2ar(CPB) 1 Further, ar(PBQ) = 2 ar(PBC) [PQ is median] .. ar(ARC) = ar(PBQ).