D and E are the points on the sides AB and AC respectively of a \[\Delta \,ABC\] and \[AD=8\,cm,\] \[DB=12\,cm,\] \[AE=6\,cm.\] and \[EC=9\,cm,~\] then BC is equal to
A) \[\frac{2}{5}DE\]
B) \[\frac{5}{2}DE\]
C) \[\frac{3}{2}DE\]
D) \[\frac{2}{3}DE\]
Correct Answer:
B
Solution :
As in \[\Delta ADE\] and \[\Delta ABC\] \[\frac{AD}{AB}=\frac{8}{20}=\frac{2}{5},\frac{AE}{EC}=\frac{6}{15}=\frac{2}{5}\] So, \[\frac{AD}{AB}=\frac{AE}{EC}\] And \[\angle A=\angle A\] (Common) \[\Delta ADE\sim \Delta ABC\] \[\therefore \] \[\frac{DE}{BC}=\frac{AD}{AB}\] \[\Rightarrow \] \[\frac{DE}{BC}=\frac{2}{5}\] \[\Rightarrow \] \[BC=\frac{5}{2}DE\]