A triangle has corners at #(3 , 2 )#, #(1 ,7 )#, and #(5 ,4 )#. What is the radius of the triangle's inscribed circle?

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Answer 1 · 2017-07-15T09:28:03

The radius of the incircle is #=1.06#

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The length of the sides of the triangle are

#c=sqrt((1-3)^2+(7-2)^2)=sqrt(4+25)=sqrt29=5.39#

#a=sqrt((5-1)^2+(4-7)^2)=sqrt(16+9)=5#

#b=sqrt((5-3)^2+(4-2)^2)=sqrt(4+4)=sqrt8=2.83#

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|(3,2,1),(1,7,1),(5,4,1)|#

#=1/2(3*|(7,1),(4,1)|-2*|(1,1),(5,1)|+1*|(1,7),(5,4)|)#

#=1/2(3(7-4)-2(1-5)+1(4-35))#

#=1/2(9+8-31)#

#=1/2|-14|=7#

The radius of the incircle is #=r#

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

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

#=14/(13.22)=1.06#