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

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Answer 1 · 2017-06-08T14:31:38

The radius of the inscribed circle is #=1.02#

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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#