Question

A triangle has sides with lengths of 8, 4, and 6. What is the radius of the triangle's inscribed circle?

Answer 1

We can use semiperimeter and Heron's formula

First, we find the semiperimeter, which is the sum of the sides divided by two.

Therefore, `(8+4+6)/2 = 9` which is the semiperimeter

Now, we can use Heron's formula to find the area, which is
`sqrt((s)(s-a)(s-b)(s-c)) = A`

We just plug in values now
`sqrt((9)(9-8)(9-4)(9-6))`

which is
`sqrt((9)(1)(5)(3))`
which is
`sqrt135`

Therefore, `sqrt135 = A`

Now, we can use the formula relating inradii, Area, and semiperimeter

`A = r * s`

Since we found two values, we can just plug in to find the third.
`sqrt135 = r * 9`

`sqrt(135) / 9 = r`

And we are done.