Answer:
about 4.03 units
Explanation:
The incircle of a triangle is tangent to each of the sides. The points of tangency from a given vertex are the same lengths. In a right triangle, the points of tangency from the right angle are equal in length to the radius of the incircle. These facts let us write an equation for the perimeter of the triangle in terms of the incircle radius and the side lengths:
2(r) +2(a -r) +2(b-r) = a +b +c
2a +2b -2r = a +b +c . . . . . simplify
a +b -c = 2r . . . . . . . . . . . add 2r -a -b -c
r = (a +b -c)/2 . . . . . radius of the incircle
For our triangle, with a=5, b=12, c=13, the radius is ...
r = (5 +12 -13)/2 = 2
In our diagram, if point C is the origin, the coordinates of the incenter (D) are (2, 2). The circumcenter (E) is the midpoint of the hypotenuse, so will have coordinates (b/2, a/2) = (6, 2.5).
The distance formula can be used to find the distance between D and E:
d = (√((6 -2)² +(2.5 -2)²) = √16.25
The distance between the incenter and circumcenter is √16.25 ≈ 4.0311.
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Additional comment
In the attached diagram, we used angle bisectors to locate the incenter. This was done before we realized how simple it was to calculate its location.