What is the radius of the circle inscribed in a triangle with side lengths $5$, $6$, and $9$?
(Ignore that the image says the triangle is inscribed in a circle.)
I'm not sure where to start and, as well as hints/an answer, advice on solving such geometry questions (apart from the usual, add in a few lines and see what happens!) would be much appreciated!
Thank you!
Note that the lines joining the angles to the incentre divide the triangle into three smaller triangles, with bases $a,\,b$ and $c$ respectively and each with height $r$. The sum of areas of these three triangles, hence the area $A$ of the original triangle, is $$A=\frac{ar}{2} + \frac{br}{2} + \frac{cr}{2}$$ We get the inradius as $$r=\frac{A}{s}\quad...(s=\frac{a+b+c}{2})$$