Question

In the given figure, PORST is a pentagon and QO
Il PR, which meets SR produced at O. Prove that
ar(OPTS) = ar(PQRST).​

Answer 1

Concept we will be using:

(i)Triangles on the same base and between same parallels are equal in area.

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Given: PQRST is pentagon. TX || SP And RY || SQ (Please refer to the attachment)

RTP: ar(PQRST) = ar(SXY)

Proof:

ΔSPX and ΔSPY are triangles on the same base SP and between same parallels TX and SP because TX || SP

Therefore, ar(ΔSPX) =ar(ΔSPT)

ΔSPX and ΔSPT are triangles on the same base SP and between same parallels TX and SP (Given: TX || SP)

Therefore, ar(ΔSPX) =ar(ΔSPT)

ΔSQY and ΔSQR are triangles on the same base SQ and between same parallels RY and SQ (Given: RY || SQ)

Therefore, ar(ΔSQY) =ar(ΔSQR)

And,

ar(ΔSXY)

=ar(ΔSPX) + ar(ΔSQY) + ar(ΔSPQ)

ar(PQRST)

= ar(ΔSPT) + ar(ΔSQR)+ ar(ΔSPQ)

Plug in ar(ΔSPT)=ar(ΔSPX) and ar(ΔSQR)=ar(ΔSQY):

=ar(ΔSPX) + ar(ΔSQY) + ar(ΔSPQ)

=ar(ΔSXY)

Therefore, ar(PQRST) = ar(SXY) (Proved)