A triangle has corners at $(2 , 6 )$ , $(3 ,9 )$ , and $(4 ,5 )$ . What is the radius of the triangle's inscribed circle?

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Answer 1 · 2016-05-15T09:25:20

Radius of triangle's inscribed circle is $0.6864$

If the sides of a triangle are $a$ , $b$ and $c$ , then the area of the triangle $Delta$ is given by the formula

$Delta=sqrt(s(s-a)(s-b)(s-c))$ , where $s=1/2(a+b+c)$

and radius of inscribed circle is $Delta/s$

Hence let us find the sides of triangle formed by $(2,6)$ , $(3,9)$ and $(4,5)$ . This will be surely distance between pair of points, which is

$a=sqrt((3-2)^2+(9-6)^2)=sqrt(1+9)=sqrt10=3.1623$

$b=sqrt((4-3)^2+(5-9)^2)=sqrt(1+16)=sqrt17=4.1231$ and

$c=sqrt((4-2)^2+(5-6)^2)=sqrt(4+1)=sqrt5=2.2631$

Hence $s=1/2(3.1623+4.1231+2.2631)1/2xx9.5485=4.7742$

and $Delta=sqrt(4.7742xx(4.7742-3.1623)xx(4.7742-4.1231)xx(4.7742-2.2631)$

= $sqrt(4.7742xx1.6119xx0.6511xx2.1432)=sqrt10.7386=3.277$

And radius of inscribed circle is $3.277/4.7742=0.6864$

A triangle has corners at (2 , 6 )(2 , 6 ), (3 ,9 )(3 ,9 ), and (4 ,5 )(4 ,5 ). What is the radius of the triangle's inscribed circle?