In the adjoining figure `PQ_|_BC, AD_|_BC`, then find following ratios:

(i( `(A(DeltaPQB))/(A(DeltaPBC))` (ii) `(A(DeltaPBC))/(A(DeltaABC))`
(iii) `(A(DeltaABC))/(A(DeltaADC))`(iv) `(A(DeltaADC))/(A(DeltaPQC))`

Explanation

`DeltaPQB` and `DeltaPBC` have same height `PQ`. Areas of triangles with equal heights are proportional to their corresponding bases.
`:.(A(DeltaPQB))/(A(DeltaPBC))=(QB)/(BC)`
(iii) `DeltaPBC` and `DeltaABC` have same base BC.
Area of triangles with equal bases are propoprtional to their corresponding heights.
`:.(A(DeltaPBC))/(A(DeltaABC))=(PQ)/(AD)`
(iii) `DeltaBC` and `DeltaADC` have same height `AD`. Areas of triangles with equal heights are proportional to their corresponding bases.
`:.(A(DeltaABC))/(A(DeltaADC))=(BC)/(DC)`
(iv) Ratio of areas of two triangles is equal to the ratio of the products of their bases and corresponding heights.
`:.(A(DeltaADC))/(A(DeltaPQC))=(DCxxAD)/(QCxxPQ)`
Ans.
(i) `(A(DeltaPQB))/(A(DeltaPBC))=(QB)/(BC)` (ii) `(A(DeltaPBC))/(A(DeltaABC))=(PQ)/(AD)`
(iii) `(A(DeltaABC))/(A(DeltaADC))=(BC)/(DC)` (iv) `(A(DeltaADC))/(A(DeltaPQC))=(DCxxAD)/(QCxxPQ)`