Question

A triangle has sides with lengths of 2, 8, and 8. What is the radius of the triangles inscribed circle?

Answer 1

`.882`

Explanation:

Refer to the figure below

Applying the Law of Sines
`2/sin alpha=8/sin beta`

Because it is a triangle
`alpha+beta+beta=180^@` => `alpha=180^@-2beta`

Then
`1/sin (180^@-2beta)=4/sin beta`

Sines of supplementary angles are equal or `sin (180^@-theta)=sin theta`.
So
`1/(sin 2beta)=4/sin beta`
`1/(2cancel(sin beta)*cos beta)=4/cancel(sin beta)`
`cos beta=1/8` => `beta=82.819^@`

As we can see from the figure
`tan (beta/2)=r/1` => `r=tan(beta/2)`

In case,
`r=tan(82.819^@/2)=.882`