Question

PQRS is a parallelogram,. A is any point on SR. PA is produced to meet QR produced at B . prove that area of triangle QAR =area of triangle SAB​

Answer 1

Answer:

it has been proved below...

Step-by-step explanation:

GIVEN- PQRS is a parallelogram and PS=BR.

TO PROVE- ar(QAR)=ar(SAB)

CONSTRUCTION- Join PR

PROOF- Since, triangles PAR and QAR are on the same base AR and between same parallels AR and PQ.Therefore,

ar(PAR)=ar(QAR)

Now, PQRS is a parallelogram.

=> PS=QR

But, it is given that PS=BR.SO,

=> PS||BR and PS=BR.

Thus, in quadrilateral PSBR; one pair of opposites sides is equal and parallel.

Therefore, PSBR is a parallelogram

Now, in triangles PAR and SAB, we have

PA=AB (diagonals of a ||gm bisect each other)

angle PAR= angle SAB(vertically opposite angles)

AR=SA(diagonals of a ||gm bisect each other)

So, by SAS cong. rule, we have

triangle PAR congurent to triangle SAB

But, triangle PAR= triangle QAR (proved above)

So, triangle QAR is congurent to triangle SAB

Since, congurent triangles have equal area, therefore

=> ar(QAR)=ar(SAB)

Hence, proved

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