Question : In $\triangle$ABC, the bisector of $\angle$BAC intersects BC at D and the circumcircle of $\triangle$ABC at E. If AB : AD = 3 : 5, then AE : AC is:
Option 1: 5 : 3
Option 2: 3 : 2
Option 3: 2 : 3
Option 4: 3 : 5
Correct Answer: 3 : 5
Solution :
Given:
AB : AD = 3 : 5
AE is bisector of $\angle$BAC.
So, $\angle$BAD = $\angle$CAE
$\angle$AEC = $\angle$ABC [As AC is a common segment]
$\therefore \triangle$ABD $\sim \triangle$AEC
So, $\frac{AB}{AE}=\frac{AD}{AC}$
⇒ $\frac{AB}{AD}=\frac{AE}{AC}$
$\therefore \frac{AE}{AC}=\frac{3}{5}$
Hence, the correct answer is 3 : 5.
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