A triangle has corners at #(2 ,3 )#, #(1 ,2 )#, and #(5 ,8 )#. What is the radius of the triangle's inscribed circle?

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Answer 1 · 2018-03-02T12:50:10

#radiusapprox1.8# units

Let the vertices of #DeltaABC# are #A(2,3)#, #B(1,2)# and #C(5,8)#.

Using distance formula,

#a=BC=sqrt((5-1)^2+(8-2)^2)=sqrt(2^2*13)=2*sqrt(13)#

#b=CA=sqrt((5-2)^2+(8-3)^2)=sqrt(34)#

#c=AB=sqrt((1-2)^2+(2-3)^2)=sqrt(2)#

Now, Area of #DeltaABC=1/2|(x_1,y_1,1), (x_2,y_2,1),(x_3,y_3,1)|#

#=1/2|(2,3,1), (1,2,1),(5,8,1)|=1/2|2*(2-8)+3*(1-5)+1*(8-10)|=1/2|-12-12-2|=13# sq. units

Also, #s=(a+b+c)/2=(2*sqrt(13)+sqrt(34)+sqrt(2))/2=approx7.23# units

Now, let #r# be the radius of triangle's incircle and #Delta# be the area of triangle, then

#rarrr=Delta/s=13/7.23approx1.8# units.