A triangle has corners at $(5 , 2 )$ , $(9 ,9 )$ , and $(6 ,8 )$ . What is the radius of the triangle's inscribed circle?

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Answer 1 · 2016-10-24T16:04:33

Radius of triangle's inscribed circle is $0.9733$

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 $(5,2)$ , $(9,9)$ and $(6,8)$ . This will be surely distance between pair of points, which is

$a=sqrt((9-5)^2+(9-2)^2)=sqrt(16+49)=sqrt65=8.0623$

$b=sqrt((6-9)^2+(8-9)^2)=sqrt(9+1)=sqrt10=3.1623$ and

$c=sqrt((6-5)^2+(8-2)^2)=sqrt(1+36)=sqrt37=6.0828$

Hence $s=1/2(8.0623+3.1623+6.0828)=1/2xx17.3074=8.6537$

and $Delta=sqrt(8.6537xx(8.6537-8.0623)xx(8.6537-3.1623)xx(8.6537-6.0828)$

= $sqrt(8.6537xx0.5914xx5.3914xx2.5709)=sqrt70.9365=8.4224$

And radius of inscribed circle is $8.4224/8.6537=0.9733$

A triangle has corners at (5 , 2 )(5 , 2 ), (9 ,9 )(9 ,9 ), and (6 ,8 )(6 ,8 ). What is the radius of the triangle's inscribed circle?