14. find the coordinates of the circumcenter for ∆def with coordinates d(1,3) e (8,3) and f(1,-5). show your work.

As in the solution of all geometry problems, the first step is to draw a diagram.  In this case, we see a right-triangle for which finding the circumcentre is almost trivial.

Recall that the circumcentre is the centre of a circle which passes through all three points D, E and F (shown as A,B & C in the attached figure).

The circumcentre can be found by the intersection of the perpendicular bisectors of the three sides.  Since the three bisectors are concurrent, we only need the intersection of any two perpendicular bisectors out of the three.

In a right triangle where the legs are parallel to the axes, the intersection of the perpendicular bisectors can be found by inspection.

In the given case, 
mid-point of DE (shown as AB) = ((1+8)/2, 3) = (4.5,3)
   perpendicular bisector of DE :  x=4.5
mid-point of DF (shown as AC) = (1, (3-5)/2) = (1,-1)
   perpendicular bisector of DF :  y=-1

Therefore the circumcentre is the intersection of x=4.5 and y=-1, or
   circumcentre = (4.5,-1)

which brings us to the interesting property of the circumcentre of a right-triangle:  The circumcentre of a right triangle is located at the mid-point of the hypotenuse.

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