In ΔABC, D and E are points on sides AB and BC, respectively, such that DE∥AC. If BD=4 cm and AB=12 cm, then the ratio of the area of ΔBDE to that of quadrilateral ACED is:

In ΔABC, since DE∥AC, ΔBDE∼ΔBAC by AA similarity.
The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides:

Area(ΔBAC)
Area(ΔBDE)
​
=(
AB
BD
​
)
2

Given BD=4cm and AB=12cm:

Area(ΔBAC)
Area(ΔBDE)
​
=(
12
4
​
)
2
=(
3
1
​
)
2
=
9
1
​

Let the area of ΔBDE=1k and the area of ΔBAC=9k.

The area of the quadrilateral ACED is:

Area(ACED)=Area(ΔBAC)−Area(ΔBDE)
Area(ACED)=9k−1k=8k
Thus, the ratio of the area of ΔBDE to that of the quadrilateral ACED is:

Area(ACED)
Area(ΔBDE)
​
=
8k
1k
​
=1:8
(Note: Correcting option D to 1:8 matches the exact geometric derivation).