A triangle has corners at #(3 , 3 )#, #(1 ,2 )#, and #(5 ,4 )#. What is the radius of the triangle's inscribed circle?

1 Answer
Shwetank Mauria
May 15, 2016

The three points lie on same straight line and hence no distinct triangle can be formed. In other words, one could say that radius of inscribed circle is #0#.

Explanation:

If the sides of a triangle are #a#, #b# and #c#, then the area of the triangle #Delta# is given by the formula

#Delta=sqrt(s(s-a)(s-b)(s-c))#, where #s=1/2(a+b+c)#

and radius of inscribed circle is #Delta/s#

Hence let us find the sides of triangle formed by #(3,3)#, #(1,2)# and #(5,4)#. This will be surely distance between pair of points, which is

#a=sqrt((1-3)^2+(2-3)^2)=sqrt(4+1)=sqrt5=2.236#

#b=sqrt((5-1)^2+(4-2)^2)=sqrt(16+4)=sqrt20=4.472# and

#c=sqrt((5-3)^2+(4-3)^2)=sqrt(4+1)=sqrt5=2.236#

As #a+c=b#, it is apparent that the three points lie on same straight line and hence no distinct triangle can be formed.

In other words, one could say that radius of inscribed circle is #0#.

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