Question

In ∆PQR three medians intersect at S prove that ar(∆PSR)=ar(∆QSR)=ar(∆PSQ)=1/3ar(∆PQR)

Answer 1

Answer: The answer is explained in the picture.

Answer 2

ar(∆PSR)=ar(∆QSR)=ar(∆PSQ)=1/3ar(∆PQR) if three median intersect at S

Step-by-step explanation:

Let say Median are PM , QN & RO in ∆PQR

Median divides triangle into two Equal Areas

=> Area of Δ PQM =  Area of Δ PRM = (1/2)  Area of Δ PQR

   Area of Δ QPN =  Area of Δ QRN = (1/2)  Area of Δ PQR

     Area of Δ RPO =  Area of Δ RQO = (1/2)  Area of Δ PQR

Area of Δ PQM =  Area of Δ PRM =    Area of Δ QPN =  Area of Δ QRN =   Area of Δ RPO =  Area of Δ RQO

Area of Δ PQM  =  Area of Δ PQS  + Area of Δ QSM

Area of Δ PQN  =  Area of Δ PQS  + Area of Δ PSN

as Area of Δ PQM  = Area of Δ PQN

=> Area of Δ PQS  + Area of Δ QSM = Area of Δ PQS  + Area of Δ PSN

=>  Area of Δ QSM =  Area of Δ PSN

Similarly we can show that

Area of Δ QSM =  Area of Δ PSN = Area of Δ RSM  = Area of Δ RSN = Area of Δ PSO = Area of Δ QSO

Area of Δ QSM +  Area of Δ PSN + Area of Δ RSM  + Area of Δ RSN + Area of Δ PSO + Area of Δ QSO = Area of Δ PQR

=> Area of Δ QSM =  Area of Δ PSN = Area of Δ RSM  = Area of Δ PSN = Area of Δ PSO = Area of Δ QSO =  Area of Δ PQR / 6

ar(∆PSR) = Area of Δ PSN +Area of Δ RSN

=> ar(∆PSR) =  Area of Δ PQR / 6 +  Area of Δ PQR / 6

=>  ar(∆PSR) = Area of Δ PQR / 3

Similalrly ar(∆QSR)=ar(∆PSQ) = Area of Δ PQR / 3

Hence ar(∆PSR)=ar(∆QSR)=ar(∆PSQ)=1/3ar(∆PQR)

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