A triangle has corners at #(2 , 2 )#, #(1 ,3 )#, and #(6 ,4 )#. What is the radius of the triangle's inscribed circle?

1 Answer
Burglar
May 29, 2016

The radius is approximately 0.54.

Explanation:

There is a formula to calculate the radius of inscribed circle based on the perimeter and area of the triangle.
If the area is #A# and the perimeter is #P# the radius of the circle is #r=2A/P#.
So our goal is to find area and perimeter.

First the perimeter, it is given by the distance of the points in pairs.
Lets call the three points #p_1=(2,2)#, #p_2=(1,3)#, #p_3=(6,4)# the length of the three sides are

side between #p_1# and #p_2# is
#s_1 = d(p_1, p_2)=sqrt((2-1)^2+(2-3)^2)=sqrt(2)\approx1.41#

side between #p_2# and #p_3# is
#s_2 = d(p_3, p_3)=sqrt((1-6)^2+(3-4)^2)=sqrt(26)\approx5.1#

side between #p_3# and #p_1# is
#s_3 = d(p_3, p_1)=sqrt((6-2)^2+(4-2)^2)=sqrt(20)\approx4.47#

The perimeter is then #P=s_1 + s_2 + s_3 = 1.41+5.1+4.47=10.98#
The Area can be calculated with the formula
#A=sqrt(P/2(P/2-s_1)(P/2-s_2)(P/2-s_3))#
#=sqrt(5.49(5.49-1.41)(5.49-5.1)(5.49-4.47))#
#=sqrt(5.49*4.08*0.39*1.02)#
#=sqrt(8.91)#
#A\approx2.98#.

Now it is time for the initial formula

#r=2A/P=2*2.98/10.98\approx0.54#.

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