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 #9
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Apr 25, 2020
Apr 24, 2020
 #2
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1)  I'm going to put this on a coordinate axis.

     I place point A at the origin:  A = (0,0)

     I place point C at the point (16,0)

     Since this is an isosceles triangle, point B will be somewhere above the midpoint of AC.

     The distance from A to this midpoint is 8; the distance from A to point B is 17.

     This makes an 8 - 15 - 17 right triangle, so point B = (8,15)

     The circumcircle has its center at the intersection points of the perpendicular bisectors of the sides. 

     The line drawn from point B to the midpoint of AC is one of these perpendicular bisectors and its equation is x = 8.

     Now, I plan to find the perpendicular bisector of AB and find where this line crosses the line x = 8.

     The slope of AB is:  (15 - 0) / (8 - 0)  =  15/8.

     This means that the perpendicular bisector of AB has a slope of -8/15.

     The midpoint of AB is the point (4, 7.5).

     The equation of the perpendicular bisector of AB is:  y - 7.5  =  (-8/15)(x - 4)

      Multiplying this out:  15y - 112.5  =  -8x + 32     --->     8x + 15y  =  144.5.

      Now, to find where this line intersects the line x = 8:    8(8) + 15y  =  144.5   --->   15y  =  80.5

                     --->   y  =  5.03125

      This means that the circumcenter is (8, 5.03125)

       To find the radius, I will find the distance from this point to the point (0,0):

            distance  =  sqrt( (8.5 - 0)2 + (5.03125 - 0)2 )  =  sqrt( 97.56347656 )  =  9.877422567.

 

I hope that this is clear and I hope that I didn't mess it up!

Apr 24, 2020

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