A triangle has corners at #(8 , 9 )#, #(7 ,3 )#, and #(2 ,4 )#. What is the radius of the triangle's inscribed circle?

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Answer 1 · 2017-06-28T07:04:47

The radius of the incircle is #=1.63#

The area of the triangle is

#A=1/2|(x_1,y_1,1),(x_2,y_2,1),(x_3,y_3,1)|#

#A=1/2|(8,9,1),(7,3,1),(2,4,1)|#

#=1/2(8*|(3,1),(4,1)|-9*|(7,1),(2,1)|+1*|(7,3),(2,4)|)#

#=1/2(8(3-4)-9(7-2)+1(28-6))#

#=1/2(-8-45+22)#

#=1/2|-31|=31/2#

The length of the sides of the triangle are

#a=sqrt((8-7)^2+(9-3)^2)=sqrt(37)#

#b=sqrt((7-2)^2+(3-4)^2)=sqrt26#

#c=sqrt((8-2)^2+(9-4)^2)=sqrt61#

Let the radius of the incircle be #=r#

Then,

The area of the circle is

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

The radius of the incircle is

#r=(2a)/(a+b+c)#

#=(31)/(sqrt37+sqrt26+sqrt61)#

#=31/18.99=1.63#