question_answer

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)             \[\Delta ADE\sim \Delta ABC\] \[\therefore \]      \[\frac{DE}{BC}=\frac{AD}{AB}\] \[\Rightarrow \] \[\frac{DE}{BC}=\frac{2}{5}\] \[\Rightarrow \]   \[BC=\frac{5}{2}DE\]