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

1 Answer
Oct 2, 2016

Inradius#=0.5097#

Explanation:

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The radius of a triangle's inscribed circle is called inradius.

Inradius formula : #r=2A/p#, where #A# = Area of the triangle and #p#=perimeter of the triangle.

1) calculate d the distance between the corners of the triangles, use the distance formula :
#d=sqrt((x2−x1)^2+(y2−y1)^2#

Given #A(3,3), B(1,2),C(2,1)#
#=> AB=sqrt((1-3)^2+(2-3)^2)=sqrt(2^2+1^2)=sqrt5#
#=> AC=sqrt((1-3)^2+(2-3)^2)=sqrt(2^2+1^2)=sqrt5#
#=> BC=sqrt((2-1)^2+(1-2)^2)=sqrt(1^2+1^2)=sqrt2#

perimeter #p=AB+AC+BC=sqrt5+sqrt5+sqrt2=2sqrt5+sqrt2#

Since #AB=AC#, the triangle is isosceles.

Since triangle ABD is right-angled,
#=> AD=sqrt((AB^2-(BD)^2)#; #(BD=(1/2)BC)#
#=> AD=sqrt((sqrt5)^2-(sqrt(2)/2)^2)=sqrt(5-2/4)=sqrt(9/2)=3/sqrt2#

Area of triangle #A= (1/2)*AD*BC=(1/2)*(3/sqrt2)*sqrt2=3/2#

Inradius #r=(2A)/p=(2*(3/2))/(2sqrt5+sqrt2)=0.5097#