A triangle has corners at #(3 , 4 )#, #(8 ,2 )#, and #(1 ,8 )#. What is the radius of the triangle's inscribed circle?

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Answer 1 · 2016-05-14T04:59:47

The radius of inscribed circle #r=0.8387131004" "#unit

Let us label the points #A(3, 4)#, #B(8, 2)#, #C(1, 8)#
Solve for the distances #a=BC# and #b=AC# and #c=AB#

The radius of the inscribed circle #r# formula

#r=sqrt(((s-a)(s-b)(s-c))/s)#

Solve for the values of s, a, b, and c first

#a=sqrt((x_b-x_c)^2+(y_b-y_c)^2)#
#a=sqrt((8-1)^2+(2-8)^2)#
#a=sqrt85#

#b=sqrt((x_a-x_c)^2+(y_a-y_c)^2)#
#b=sqrt((3-1)^2+(4-8)^2)#
#b=sqrt20#

#c=sqrt((x_a-x_b)^2+(y_a-y_b)^2)#
#c=sqrt((3-8)^2+(4-2)^2)#
#c=sqrt(29)#

Half the Perimeter of the triangle #s#

#s=(a+b+c)/2=(sqrt85+sqrt20+sqrt29)/2#

Compute #r#

#r=#
#sqrt((((sqrt85+sqrt20+sqrt29)/2-a)((sqrt85+sqrt20+sqrt29)/2-b)((sqrt85+sqrt20+sqrt29)/2-c))/[0.5*(sqrt85+sqrt20+sqrt29)])#

#r=0.8387131004#

God bless....I hope the explanation is useful.