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

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Answer 1 · 2017-01-12T19:26:43

#0.5402 (4dp)#.

Let us name the given points #A(2,1), B(3,4) and C(1,2)#.

Using Distance Formula, we find that,

#AB^2=(2-3)^2+(1-4)^2=1+9=10#

Similarly, #BC^2=8, and, AC^2=2#.

We notice that, #BC^2+AC^2=AB^2,# which means that

#Delta ABC# is right-angled having #AB# as its Hypotenuse.

We know from Geometry that, the Inradius #r# of a right #Delta# is

#1/2#(sum of the lengths of sides making the right angle-length of

hypo.)

#:. r=1/2[sqrt2+sqrt8-sqrt10]=1/2(3sqrt2-sqrt10)#

#=sqrt2/2(3-sqrt5)~~1.4142/2(3-2.2361)~~0.5402(4 dp)#.