Question

In the given figure, $\angle AMN=\angle MBC={76}^o$. If a, b and c are the lengths of AM, MB and BC respectively then express the length of MN in terms of a, b and c.

Answer 1

$In△AMNand△ABC,\angle AMN=\angle MBC\left(given\right)\angle MAN=\angle BAC\left(Common\right)\angle ANM=\angle BCA\left(Anglesumproperty\right)$

$\therefore △AMN\sim △ABC\Rightarrow \frac{AM}{AB}=\frac{MN}{BC}\Rightarrow \frac{a}{a+b}=\frac{MN}{c}\Rightarrow MN=\frac{ac}{a+b}$