Question

If the lengths of the sides BC, CA and AB of a $\Delta ABC$ are a, b and c respectively and AD is the bisector of $\angle A$ then find the lengths of BD and DC.

Answer 1

It is given that AB = c, BC = a and CA = b

We need to find out, BD and DC.

Since, AD is the bisector of $\angle A$

$\frac{AB}{AC}=\frac{BD}{CD}$
$\frac{c}{b}=\frac{BD}{(BC-BD)}$
$\frac{c}{b}=\frac{BD}{a-BD}$
$c(a-BD)=b\left(BD\right)$
$ac=(b+c)BD$
$BD=\frac{ac}{(b+c)}$

In ΔABC, AD is the bisector of $\angle A$ , meeting side BC at D.

We need to find DC,

Since, AD is $\angle A$ bisector,

Then,
$\frac{AB}{AC}=\frac{BD}{DC}$
$\frac{c}{b}=\frac{\left[\frac{ac}{b+c}\right]}{DC}$
$DC=\frac{b\left(ac\right)}{c(b+c)}$
$DC=\frac{ab}{b+c}$