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 Tier II, 2015]

A) 8

B) 4

C) 3

D) 6

Correct Answer: D

Solution :

[d]                         AD extended meets BC at F. \[\angle \,ADB=\angle \,BDF=90{}^\circ \] \[\angle \,ADB=\angle \,FDB\](BD is the angle bisector) \[\therefore \]      \[\angle \,BAD=\angle \,BFD\] \[\Rightarrow \]   \[\Delta \,ABD\] and \[\Delta \,FBD\]are congruent. \[\Rightarrow \]   AD = DF and  \[\Delta \,ADE\]is similar to \[\Delta \,AFC\]\[(\therefore DE\parallel BC)\] \[\frac{AE}{AC}=\frac{AD}{AF}=\frac{1}{2}\]\[\Rightarrow \]\[AE=\frac{1}{2}\times 12=6\,cm\]