A triangle has corners at #(5 ,1 )#, #(7 ,6 )#, and #(2 ,3 )#. What is the radius of the triangle's inscribed circle?

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Answer 1 · 2017-07-05T07:31:48

The radius of the incircle is #=1.28#

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|(5,1,1),(7,6,1),(2,3,1)|#

#=1/2(5*|(6,1),(3,1)|-1*|(7,1),(2,1)|+1*|(7,6),(2,3)|)#

#=1/2(5(6-3)-1(7-2)+1(21-12))#

#=1/2(15-5+9)#

#=1/2|19|=19/2#

The length of the sides of the triangle are

#a=sqrt((7-5)^2+(6-1)^2)=sqrt(29)#

#b=sqrt((7-2)^2+(6-3)^2)=sqrt34#

#c=sqrt((5-2)^2+(1-3)^2)=sqrt13#

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)#

#=(19)/(sqrt29+sqrt34+sqrt13)#

#=19/14.82=1.28#