In the given figure, AP || BQ || CR. Prove that ar (AQC) = ar (PBR).

detailed solution

1

Since ∆ABQ and ∆PBQ lie on the same base BQ and are between the same parallels AP and BQ,

According to Theorem 9.2: Two triangles on the same base (or equal bases) and between the same parallels are equal in area.

Area (ΔABQ) = Area (ΔPBQ) ...(1)

Similarly, ∆BCQ and ∆BRQ lie on the same base BQ and are between the same parallels BQ and CR.

Area (ΔBCQ) = Area (ΔBRQ) ... (2)

On adding Equations (1) and (2), we obtain

Area (ΔABQ) + Area (ΔBCQ) = Area (ΔPBQ) + Area (ΔBRQ)

Hence, Area (ΔAQC) = Area (ΔPBR) is proved.