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