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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