Question

If the length of the sides BC, CA and AB of a △ABC are a, b and c respectively and AD is the bisector of ∠A then find the length of BD and DC.

Answer 1

Let DC = x
∴ BD = a − x
By using angle bisector theore in △ABC, we have
$$\begin{gathered} \frac{\mathrm{AB}}{\mathrm{AC}}=\frac{\mathrm{BD}}{\mathrm{DC}} \\ \Rightarrow \frac{c}{b}=\frac{a-x}{x} \\ \Rightarrow cx=ab-bx \\ \Rightarrow x\left(b+c\right)=ab \\ \Rightarrow x=\frac{ab}{\left(b+c\right)} \end{gathered}$$
Now,
$$\begin{gathered} a-x=a-\frac{ab}{b+c} \\ =\frac{ab+ac-ab}{b+c} \\ =\frac{ac}{\left(a+b\right)} \end{gathered}$$