Question

In the given figure, TQ || SR and PQ || TS. Prove that area (ΔPTS) = area (ΔTRQ).

[Hint: Join QS.]

Answer 1

Given:

TQ || SR

PQ || TS

To prove: Area(△PTS) = Area(△TRQ)

Step 1: Join QS

By joining QS, we divide the quadrilateral PTSQ into two triangles:

△PQS and △PTS

Similarly, quadrilateral TRSQ is divided into:

△QRS and △TRQ

Step 2: Use parallel line properties

Since PQ || TS, quadrilateral PTSQ is a parallelogram.

Thus,

Area(△PQS) = Area(△PTS)

Similarly, since TQ || SR, quadrilateral TRSQ is also a parallelogram.

Thus, 

Area(△QRS) = Area(△TRQ)

Step 3: Relating areas

Notice that triangles △PQS and △QRS lie on the same base QS and between the same parallels PR and QS

Area(△PQS) = Area(△QRS)

Step 4: Combine

From Step 2 and Step 3:

Area(△PTS) = Area(△PQS) 

Area(△TRQ) = Area(△QRS)

And since Area(△PQS) = Area(△QRS)

Area(△PTS) = Area(△TRQ)