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Theorem 9.2: Two triangles on the same base (or equal bases) and between thesame parallels are equal in area. |
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Answer» Triangles on the same base and between same parallels are equal in area. Given: Two triangles ABC and DBC are on the same base BC and between same parallels EF and BC. To Prove : ar(ΔABC) = ar(ΔDBC) Construction : Through B, draw BE || AC intersecting line AD in E and through C draw CF || BD intersecting the line DA in F. Proof : EACB and DFCB are parallelograms (Since two pairs of opposite sides are parallel) Also ||gm EACB and ||gm DFCB are on the same base BC and between same parallels EF and BC.∴ ar(||gm EACB)= ar(||gm DFCB)...(i)Now AB is the diagonal of ||gm EACB∴ ar(ΔEAB) = ar(ΔABC)∴ ar(ΔABC) = (1/2)ar(||gm EACB) .....(ii)Similarly,ar(ΔDBC) = (1/2) ar(||gm DFCB)...(iii)From equations (i), (ii) and (iii), we getar(Δ ABC) = ar(Δ DBC) |
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