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A triangle with sides of 5, 12, and 13 has both an inscribed and a circumscribed circle. What is the distance between the centers of those circles?

 Nov 17, 2019
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Let A  = (0,0)

Let B  = (0,5)

Let C  = (12,0)

 

AB = 5

AC = 12

BC = 13

 

We can calculate the incenter   as  

 

x =      (0)(BC)  + (0) (AC)  + (12)(AB)               12 * 5                              

          ________________________  =           ______   =  2

                     perimeter                                        30

 

 

y =  (0)(BC)  + (5)(AC) + (0)(AB)                      5 * 12

       _________________________  =       __________   =    2

                    perimeter                                        30

 

 

So....the incenter  is  (2, 2)

 

The circumcenter  occurs at the midpoint of the hypotenuse =  [ (12 + 0) / 2 , (5 + 0) /2 ] =

  ( 6, 5/2)  = (6, 2.5)

 

The distance  between these points is

 

sqrt [ ( 6 - 2)^2  + ( 2.5 - 2)^2 ]   = sqrt [ 16 + .25 ] = sqrt [16 + 1/4]  =  sqrt [65] / 2

 

 

 

cool cool cool

 Nov 17, 2019

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