In figure, if DE`abs()`BC, then find the ratio of ar (`DeltaADE`) and ar(DECB).

Options

A. `1:3`
B. `1:2`
C. `2:5`
D. `3:5`

Answer

A

Explanation

Given, DE`abs()`BC, DE= 6 cm and BC= 12 cm
In `DeltaADE,`
`angleABC=angle(ADE)` [corresponding angle]
`angleACB=angleAED` [corresponding angle]
and `angleA=angleA` [common sides]
` therefore DeltaABC~DeltaAED` [by AAA similarity criterion]
Then, `(ar(DeltaADE))/(ar(DeltaABC))=((DE)^(2))/((BC)^(2))`
`=((6)^(2))/((12)^(2))=(1/2)^(2)`
`rArr (ar(DeltaADE))/(ar(DeltaABC))=(1/2)^(2)=1/4`
Let ar(`DeltaADE`)=k,then ar(`DeltaABC`)=4k
Now, ar(DECB)=ar(ABC)-ar(ADE)=4k-k=3k
`therefore` Required ratio=ar(ADE) : ar(DECB)=K : 3k=1 : 3`