Given the coordinates of three vertices of #DeltaABC# are
#"Corodinate of A " x_A=5,y_A=8#
#"Corodinate of B " x_B=2,y_B=8#
#"Corodinate of C " x_C=3,y_C=1#
Lengths of three sides
#AB=sqrt((x_A-x_B)^2+(y_A-y_B)^2)#
#=sqrt((5-2)^2+(8-3)^2)=sqrt34#
#BC=sqrt((x_B-x_C)^2+(y_B-y_C)^2)#
#=sqrt((2-3)^2+(3-1)^2)=sqrt5#
#CA=sqrt((x_C-x_A)^2+(y_C-y_A)^2)#
#=sqrt((3-5)^2+(1-8)^2)=sqrt53#
Now area of #DeltaABC#
#=|1/2(y_A(x_B-x_C)+y_B(x_C-x_A)+y_C(x_A-x_B))|#
#=|1/2(8(2-3)+3(3-5)+1(5-2))|#
#=|1/2(-8-6+3)|=5.5#
If the radius of the incircle of #DeltaABC " "be" "r# then
#1/2(AB+BC+CA)xxr="area "DeltaABC#
#=>1/2(sqrt34+sqrt5+sqrt53)*r=5.5#
#=>r=(2xx5.5)/(sqrt34+sqrt5+sqrt53)#
#:.r= 0.716#