question_answer

AD is perpendicular to the internal bisector of \[\angle ABC\] of \[\Delta ABC.\] DE is drawn through D and parallel to BC to meet AC at E. If the length of AC is 12 cm, then the length of AE (in cm) is               [SSC (CGL) Mains 2015]

A) 4                                 

B) 8

C) 3                                 

D) 6

Correct Answer: D

Solution :

Let AD meet BC at F.

\[\angle ADB=\angle BDF=90{}^\circ \]

As BD is angle bisector.

Let                    \[\angle DBF=\angle ABD=x\]

So,                   \[\angle BAD=\angle BFD\]

\[\therefore \] \[\Delta ABD\] and \[\Delta FBD\]are congruent

Now, \[\Delta ADE\] is similar to\[\Delta AFC.\]       \[[\because DE\parallel BC]\]

\[\therefore \]      \[\frac{AE}{AC}=\frac{AD}{AF}=\frac{1}{2}\]

\[\Rightarrow \]   \[AE=\frac{1}{2}\,\,AC=\frac{1}{2}\times 12=6\,cm\]