Question

The bisector of the angle N of a triangle MNP divides the side [MPI into segments whose lengths are 28 and 12. Find the perimeter of the triangle MNP if |MN| - |NP| = 18.

Answer 1

We know that angle bisector divides the opposite side into ratio proportional to remaining adjacent sides.

We have $\frac{MQ}{QP}=\frac{MN}{NP}$

$\therefore \frac{28}{12}=\frac{MN}{NP}$

$\therefore 28NP=12MN$ _____ (1)

But $MN-NP=18$

$\therefore MN-\frac{12}{28}MN=18$

$\therefore \frac{16}{28}MN=18\Rightarrow MN=\frac{63}{2}=31.5$

$\therefore NP=31.5-18=13.5$

Perimeter $=28+12+31.5+13.5=85$