In triangle PQR point T is the mid point of median PS as shown in the given fig show that ar(triangle PQT)=1/4 ar(triangle PQR)

in the given figure ABC is a triangle with T the midpoint of median PS 
To prove : ar (triangle PQT) = 1/4 ar(triangle PQR)
construction: join TR
​ar(triangle PQS) = ar(triangle PSR)
therefore, half of Ar(triangle PQS) = half of Ar ( triangle PSR)
that is Ar(triangle PQT)=Ar(triangle PTR)...........(1)
and Ar(triangle QST) = Ar( triangle TSR)...........(2)

since,  Ar (triangle PQR) =ar (triangles  PQS + PSR )
but from (1) and (2)
ar (triangle PQR) = 2ar(trianlge PQS)
ar (triangle PQR) =2ar(2PQT)....from (1)
ar (triangle PQR) = 4 ar (triangle PQT )
therefore,ar(triangle PQT) = 1/4 ar (triangle PQR)


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