A triangle has corners at #(3 , 4 )#, #(6 ,7 )#, and #(5 ,8 )#. What is the radius of the triangle's inscribed circle?

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Answer 1 · 2018-04-28T18:05:44

# 2sqrt2-sqrt5#.

Let the vertices be #A(3,4), B(6,7) and C(5,8)#.

Then, #AB^2=(3-6)^2+(4-7)^2=9+9=18#, & likewise,

#BC^2=2 and CA^2=20#.

#:. AB^2+BC^2=CA^2#.

Hence, #DeltaABC# is right-angled at #B#.

From Geometry, we know that the inradius #r# is,

# r=(AB+BC-CA)/2#.

#=1/2(3sqrt2+sqrt2-2sqrt5)#.

#:. r=2sqrt2-sqrt5#.