A triangle has sides with lengths of 4, 9, and 8. What is the radius of the triangles inscribed circle?

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Answer 1 · 2016-02-09T20:54:00

$1.524$

Refer to the figure below

I created this figure using MS Excel

As the sides of the triangle are 4, 8 and 9:
$x+y=9$
$x+z=8$
$y+z=4$ => $z=4-y$
$-> x+4-y=8$ => $x-y=4$

Adding the first and last equations
$2x=13$ => $x=6.5$

Using the Law of Cosines ( $triangle_(ABC)$ ):
$4^2=9^2+8^2-2*9*8*cos alpha$

$cos alpha=(81+64-16)/144=129/144$

$alpha=26.384^@$

In the right triangle with $x$ as cathetus, we can see that
$tan (alpha/2)=r/x$

$r=6.5*tan (26.384^@/2)$ => $r=1.524$