Question

A triangle has corners at `(4 , 5 )`, `(3 ,2 )`, and `(5 ,3 )`. What is the radius of the triangle's inscribed circle?

Answer 1

The radius `r=0.654929" "`unit

Explanation:

We use the formula for radius of inscribed circle in a triangle

`r=sqrt(((s-a)(s-b)(s-c))/s)" "` where a,b,c are the sides of the triangle and `s` is half of the Perimeter's measure.

`s=(a+b+c)/2`

Using distance formula we compute for the sides

Let side `a` the distance from `(4, 5)` to `(3, 2)`
Let side `b` the distance from `(5, 3)` to `(3, 2)`
Let side `c` the distance from `(4, 5)` to `(5, 3)`

`a=sqrt10`
`b=sqrt5`
`c=sqrt5`

`s=(a+b+c)/2=(sqrt10+sqrt5+sqrt5)/2=(sqrt10+2sqrt5)/2`

Compute `r`

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

`r=sqrt((((sqrt10+2sqrt5)/2-sqrt10)((sqrt10+2sqrt5)/2-sqrt5)((sqrt10+2sqrt5)/2-sqrt5))/((sqrt10+2sqrt5)/2))`

`r=0.654929" "`unit

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