Question

A triangle has sides with lengths of 3, 7, and 6. What is the radius of the triangles inscribed circle?

Answer 1

`r=sqrt(5)/2`

Explanation:

To find the radius, we use the fact that the triangle contains three internal triangles whose height is the radius `r`. The sum of the areas of these triangles equals the area of the triangle ABC, which is calculated using Heron's formula
`A = sqrt(p(p-a)(p-b)(p-c))` where `p = (a+b+c)/2`

`p=(3+7+6)/2 = 8`
`:. A= sqrt( 8*5*1*2) =sqrt(80) = sqrt(16*5) = 4sqrt(5)`
The three internal triangles have areas
`a_1 =1/2*3*r =(3r)/2`
`a_2 = 1/2*7*r =(7r)/2`
`a_3 = 1/2*6*r =3r`

`a_1 + a_2 +a_3 = A`
`r(3/2+7/2+3) = 4sqrt(5)`
`:.r = 4sqrt(5)/8 = sqrt(5)/2`