Question

The angle bisectors BD and CE of a triangle ABC are divided by the incentre I in the ratio 3:2 and 2:1. The ratio in which I divides the angle bisector through A.

• A. $3:1$
• B. $11:4$
• C. $6:5$
• D. $7:4$

Answer 1

The correct option is B $11:4$

$\because \frac{AI}{IF}=\frac{b+c}{a}$..............(1)
$\because \frac{a+c}{b}=\frac{3}{2}$................(2)
$\because \frac{CI}{IE}=\frac{a+c}{c}=\frac{2}{1}$
$\Rightarrow a+b=2c$.........(3)
(2)$2a+2c=3b$ using to
$\Rightarrow 2a+a+b=3b$ using (3)
$\Rightarrow 3a=2b$
$b=\frac{3}{2}a$.........(4)
Now again (3) $\Rightarrow$ $2c=a+b$
$=a+\frac{3}{2}a$
$\Rightarrow c+\frac{5}{4}a$
Hence $\frac{AI}{IF}=\frac{b+c}{a}=\frac{\frac{1}{2}a+\frac{5}{4}a}{a}=\frac{11}{4}$