Question

A triangle has sides with lengths of 5, 9, and 8. What is the radius of the triangles inscribed circle?

Answer 1

`1.809`

Explanation:

Refer to the figure below

As the sides of the triangle are 5, 8 and 9:
`x+y=9`
`x+z=8`
`y+z=5` => `z=5-y`
`-> x+5-y=8` => `x-y=3`

Adding the first and last equations
`2x=12` => `x=6`

Using the Law of Cosines:
`5^2=9^2+8^2-2*9*8*cos alpha`

`cos alpha=(81+64-25)/144=120/144=5/6`

`alpha=33.557^@`

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

`r=6*tan (33.557^@/2)` => `r=1.809`

Answer 2

Radius of inscribed circle is `=6/sqrt(11) ~= 1.81`

Explanation:

The radius of a circle inscribed in a triangle is
`color(white)("XXX")r= ("Area"_triangle)/s` where `s` is the semi-perimeter of the triangle.

For a triangle with sides `5, 9, and 8`
`color(white)("XXX")s=11`

Using Heron's formula
`color(white)("XXX")"Area"_triangle = sqrt(s(s-a)(s-b)(s-c))`

`color(white)("XXXXXXX")=sqrt(11(6)(2)(3)) = 6sqrt(11)`

And the required radius is
`color(white)("XXX")(6sqrt(11))/11 = 6/sqrt(11)`