Question

D, E and F are the points on sides BC, CA and AB respectively of $\Delta ABC$ such that All bisects $\angle A$, BE bisects $\angle VB$and CF bisects $\angle C$. If AB = 5 cm, BC = 8 cm and CA = 4cm determine AF. CE and BD.

Answer 1

$AF=\frac{5}{3}cm,CE=\frac{32}{13}cm,BC=\frac{40}{9}cm$
Since AD is the bisector of $\angle A$
$\therefore \frac{AB}{AC}=\frac{BD}{CD}$
$\Rightarrow \frac{5}{4}\frac{BD}{BC-BC}$

$\Rightarrow \frac{5}{4}=\frac{BD}{8-BC}$

$\Rightarrow 40-5BC=9BC=40$

$\Rightarrow BD=\frac{40}{9}cm$
Since BE is the bisector of $\angle B$
$\therefore \frac{AB}{BC}=\frac{AE}{CE}$

$\Rightarrow \frac{AB}{BC}=\frac{AC-CE}{CE}$

$\Rightarrow \frac{5}{8}=\frac{4-CE}{CE}$

$\Rightarrow 13CE=32$

$\Rightarrow \frac{32}{13}cm$
Since CF is the bisector of the $\angle C$.
$\therefore \frac{BC}{CA}=\frac{BF}{AF}$
$\Rightarrow \frac{8}{4}=\frac{AB-AF}{AF}$

$\Rightarrow 2=\frac{5-AF}{AF}$

$\Rightarrow 3AF=5$

$\Rightarrow AF=\frac{5}{3}cm$.