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

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Answer 1 · 2017-07-08T11:22:50

The radius of the incircle is #=0.44u#

The area of the triangle is

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

#=1/2(x_1(y_2-y_3)-y_1(x_2-x_3)+(x_2y_3-x_3y_2))#

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

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

#=1/2(4(2-8)-3(2-7)+1(16-14))#

#=1/2(-24+15+2)#

#=1/2|-7|=7/2#

The length of the sides of the triangle are

#a=sqrt((4-2)^2+(3-2)^2)=sqrt5#

#b=sqrt((7-2)^2+(8-2)^2)=sqrt61#

#c=sqrt((7-4)^2+(8-3)^2)=sqrt34#

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)#

#=(7)/(sqrt5+sqrt61+sqrt34)#

#=7/15.88=0.44#