Question

A triangle has corners at `(1 , 2 )`, `(5 ,7 )`, and `(9 ,5 )`. What is the radius of the triangle's inscribed circle?

Answer 1

Radius of inscribed circle `r=1.44187`

Explanation:

Let `A(x_a, y_a)=(1, 2)`
Let `B(x_b, y_b)=(5, 7)`
Let `C(x_c, y_c)=(9, 5)`

Let `a` the side from B to C
Let `b` the side from A to C
Let `c` the side from A to B

`a=sqrt((x_b-x_c)^2+(y_b-y_c)^2)`
`a=sqrt((5-9)^2+(7-5)^2)=sqrt20`

`b=sqrt((x_a-x_c)^2+(y_a-y_c)^2)`
`b=sqrt((1-9)^2+(2-5)^2)`
`b=sqrt(73)`

`c=sqrt((x_b-x_a)^2+(y_b-y_a)^2)`
`c=sqrt((5-1)^2+(7-2)^2)`
`c=sqrt(41)`

Compute half of the perimeter `s`
`s=(a+b+c)/2`
`s=9.709631968875`

Compute radius `r` of inscribed circle

`r=sqrt(((s-a)(s-b)(s-c))/s)`
`r=1.44187`

God bless....I hope the explanation is useful.