Question

AREAS OF PARALLELOGRAMS AND TRIANGLES Find the ratio of the areas of the parallelogram and the triangle lying on the same base and between the same parallels.

Answer 1

Let ΔABP and a parallelogram ABCD be on the same base AB and between the same parallels AB and PC. To Prove : ar( ΔPAB ) = (1/2)ar( ABCD) Draw BQ ||AP to obtain another parallelogram.ABQP and ABCD are on the same base AB and between the same parallels AB and PC. There fore, ar(ABQP) =  ar(ABCD) But ΔPAB ≅ ΔBQP( Diagonals PB divides parallelogram ABQP into two congruent triangles. So  ar (PAB) = ar(BQP) -----------(2) ∴ ar (PAB) = (1/2)ar(ABQP) -----------------(3) [ from (2)]This gives ar (PAB) = (1/2)ar(ABCD)   [ from (1) and (3)]