question_answer

In \[\Delta ABC,\]\[DE||BC,\]\[AD=2.5\,cm,\] \[DB=5\,cm,\]\[AE=2\,cm\]and \[BC=2\text{ }cm.\] Find EC and DE respectively. 5 cm

A)  4 cm and 3 cm

B)  5 cm and 3 cm

C)  2 cm and 4 cm

D)  4 cm and 5 cm

Correct Answer: A

Solution :

[a] In \[\Delta ADE\] and \[\Delta ABC,\]

\[\angle ADE=\angle ABC\]

[\[\because DE\parallel BC\] and AB is a transversal i.e. corresponding angle]

\[\angle AED=\angle ACB\]

[\[\because DE\parallel BC\] and AB is a transversal i.e. corresponding angle]

\[\angle A=\angle A\]                  [common]

\[\therefore \]      \[\Delta ADE\sim \Delta ABC\]

So,       \[\frac{AD}{AB}=\frac{AE}{AC}\]\[\Rightarrow \]\[\frac{2.5}{7.5}=\frac{2}{AC}\]

\[\Rightarrow \]   \[AC=\frac{2\times 75}{25}=6\]

\[\therefore \]      \[EC=AC-AE=6-2=4\,cm\]

and \[\frac{AD}{AB}=\frac{DE}{BC}\]\[\Rightarrow \]\[\frac{2.5}{7.5}=\frac{DE}{9}\]

\[\Rightarrow \]   \[DE=9\times \frac{25}{75}=3\,cm.\]