Question

A triangle has corners at `(1 , 5 )`, `(4 ,8 )`, and `(9 ,7 )`. What is the radius of the triangle's inscribed circle?

Answer 1

The radius of the inscribed circle is `=1.02`

Explanation:

Let the radius of the inscribed circle be `=r`

The area of the triangle is

`A=1/2*r*(a)+1/2*r*(b)+1/2*r*(c)`

`A=1/2r(a+b+c)`

`r=(2A)/(a+b+c)`

Let,

`A=(4,8)`

`B=(1,5)`

`C=(9,7)`

`a=sqrt((9-1)^2+(7-5)^2)=sqrt(64+4)=sqrt68`

`b=sqrt((9-4)^2+(7-8)^2)=sqrt(25+1)=sqrt26`

`c=sqrt((4-1)^2+(8-5)^2)=sqrt(9+9)=sqrt18`

so,

`(a+b+c)=sqrt68+sqrt26+sqrt18`

The area of the triangle is

`A=1/2|(4,8,1),(1,5,1),(9,7,1)|`

`=1/2(4*|(5,1),(7,1)|-8*|(1,1),(9,1)|+1*|(1,5),(9,7)|)`

`=1/2((4*-2)-(8*-8)+(1*-38))`

`=1/2(-8+64-38)`

`=9`

Therefore,

`r=(2*9)/(sqrt68+sqrt26+sqrt18)=1.02`