A triangle has corners at #(3 , 4 )#, #(5 ,6 )#, and #(2 ,1 )#. What is the radius of the triangle's inscribed circle?

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Answer 1 · 2017-06-23T08:45:46

The radius is #=0.34#

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|(3,4,1),(5,6,1),(2,1,1)|#

#=1/2(3*|(6,1),(1,1)|-4*|(5,1),(2,1)|+1*|(5,6),(2,1)|)#

#=1/2(3(6-1)-4(5-2)+1(5-12))#

#=1/2(15-12-7)#

#=1/2|-4|=2#

The length of the sides of the triangle are

#a=sqrt((5-3)^2+(6-4^2))=sqrt(8)#

#b=sqrt((5-2)^2+(6-1)^2)=sqrt34#

#c=sqrt((3-2)^2+(4-1)^2)=sqrt10#

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)#

#=(2*2)/(sqrt8+sqrt34+sqrt10)#

#=0.34#