If #A=(2,7), B=(3,1), and C=(8,9)#
then
#color(white)("XXX")abs(AB)=sqrt((3-2)^2+(1-7)^2)=sqrt(37)~~6.0828#
#color(white)("XXX")abs(BC)=sqrt((8-3)^2+(9-1)^2)=sqrt(89)~~9.4340#
#color(white)("XXX")abs(CA)=sqrt((2-8)^2+(7-9)^2)=sqrt(40)~~6.3246#
The perimeter,#p_triangle#m of the triangle is
#color(white)("XXX")"p_triangle=abs(AB)+abs(BC)+abs(CA)~~21.8413#
The semi-perimeter< #s_triangle#, of the triangle is
#color(white)("XXX")s_triangle = (p_triangle)/2~~10.9207#
The area of the triangle, #"area"_triangle# can be calculated geometrically or using Heron's Formula:
#color(white)("XXX")"area"_triangle = sqrt(s(s-a)(s-b)(s-c))~~19#
The radius, #r_circ# of a circle inscribed in a triangle is given by the formula
#color(white)("XXX")r_circ=("area"_triangle)/s ~~ 19/10.9207 ~~1.7398#
The area, #"area"_circ#, of a circle inscribed in a triangle is given by the formula
#color(white)("XXX")"area"_circ = pi * r_circ ^2 ~~9.5096#
