Question

prove that two triangle on the same base and the same parralles are equal in area

Answer 1

SOLUTION

Let ∆PQR & ∆SQR be on the same base QR and between the same parallels PS and QR.

TO PROVE : ar(∆PQR) = ar(∆SQR)

CONSTRUCTION : Draw AR || PQ & BR || SQ. We get two parallelograms PQRA & SQRB.

PROOF :  

Here ,parallelograms PQRA & SQRB lie on the same base QR and between the same parallels QR and PB.

ar (||gm PQRA) = ar (||gm SQRB).........(1)

Now in parallelogram PQRA, diagonals PR divides it into two congruent triangles.

∆PQR ≅ ∆PRA

so, ar(∆PQR ) = ar (∆PRA)

[Congruent triangles have equal areas]

ar(∆PQR ) = ½ ar (||gm PQRA).......(2)

Similarly, in parallelogram SQRB, diagonal SR divides it into two congruent triangles.

∆SQR =≈∆SRB

so, ar(∆SQR ) = ar (∆SRB)

[Congruent triangles have equal areas]

ar(∆SQR ) = ½ ar (||gm SQRB).......(3)

from eq. 1 , 2 and 3,  

ar(∆PQR ) =  ar(∆SQR )

Hence, the triangles on the same base and between same parallels are equal in area.

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