SSC
Quantitative Aptitude
Geometry
Question Bank
Triangles and Their Properties (II)
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 \[\Delta \,AFC\]\[(\therefore DE\parallel BC)\] \[\frac{AE}{AC}=\frac{AD}{AF}=\frac{1}{2}\]\[\Rightarrow \]\[AE=\frac{1}{2}\times 12=6\,cm\]