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

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Answer 1 · 2017-07-21T07:21:53

The radius of the incircle is #=0.81u#

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The corners are

#A=(4,5)#

#B=(1,3)#

#C=(5,3)#

The length of the sides of the triangle are

#c=sqrt((1-4)^2+(3-5)^2)=sqrt(9+4)=sqrt13=3.61#

#a=sqrt((5-1)^2+(3-3)^2)=sqrt(16+0)=sqrt14=4#

#b=sqrt((5-4)^2+(3-5)^2)=sqrt(1+4)=sqrt5=2.24#

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

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

#=1/2(4(3-3)-4(1-5)+1(3-15))#

#=1/2(0+20-12)#

#=1/2|8|=4#

The radius of the incircle is #=r#

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

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

#=8/(9.85)=0.81#