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# 2013 NS 30

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1 Answer: 85/8

Mar 19, 2019

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See the following image : Let A = (0, 0)   B = (10, 0)

We can find C thusly

Construct a circle with a radius of 21 centered at  A

The equation of this circle  =  x^2 + y^2 = 441     (1)

Construct anothe circle with a radius of 17 centered at  B

The equation of the circle is  (x - 10)^2 + y^2 = 289     (2)

Subtract   (2) from (1)    and we get that

x^2 - (x - 10)^2 = 152

x^2 - x^2 + 20x  - 100 = 152

20x - 252 = 0

20x = 252

x = 12.6  this is the x coordinate of C

And taking the positive  value for y

y =  sqrt (441 - (12.6)^2)  = 16.8

So....C = ( 12.6 , 16.8)

The midpoint of AC  = ( 6.3, 8.4)

The slope of the line between A and C is  8.4 /6.3 =  4/3

So...the equation of the perpendicular bisector to AC =

y = (-3/4) (x - 6.3) + 8.4

And the midpoint of AB  = (5, 0)

So....the equation of the perpendicular bisector to AB = x = 5

This is the x coordinate of the center of the circumscribing circle

The y value is      -.75 ( 5 - 6.3) + 8.4  = 9.375  = 9 + 3/8  =  75/8

So...the radius of the circumscribing circle is the distance from this point to A  ans is given by :

sqrt  (5^2 + (75/8)^2 )  =  sqrt  ( 1600 + 5625) / 8  =  sqrt (7225) / 8  =  85 / 8   Mar 19, 2019