SOLUTION: Find the center and radius of a circle circumscribing a triangle with the vertices (-2,-4) (-2,3), (5,2)

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Question 1086317: Find the center and radius of a circle circumscribing a triangle with the vertices (-2,-4) (-2,3), (5,2)
Found 2 solutions by Fombitz, htmentor:
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
So let (h,k) be the center of the circle.
Using the first and second points,
%28h%2B2%29%5E2%2B%28k%2B4%29%5E2=R%5E2=%28h%2B2%29%5E2%2B%28k-3%29%5E2
%28k%2B4%29%5E2=%28k-3%29%5E2
k%5E2%2B8k%2B16=k%5E2-6k%2B9
14k=-7
k=-1%2F2
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Using the second and third points,
%28h%2B2%29%5E2%2B+%28-1%2F2-3%29%5E2+=%28h-5%29%5E2%2B+%28-1%2F2-2%29%5E2+
h%5E2%2B4h%2B4%2B49%2F4=h%5E2-10h%2B25%2B25%2F4
14h=+25%2B25%2F4-49%2F4-4+
14h=+15+
h=15%2F14
Now that you have the center, find the distance between the center and any of the points.
That'll be the radius of the circle.
R%5E2=%285-15%2F14%29%5E2%2B%282-%28-1%2F2%29%29%5E2
R%5E2=%28%2870-15%29%2F14%29%5E2%2B%285%2F2%29%5E2
R%5E2=3025%2F196%2B25%2F4
R%5E2=3025%2F196%2B1225%2F196
R%5E2=4250%2F196
R%5E2=+2125%2F98+
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%28x-15%2F14%29%5E2%2B%28y%2B1%2F2%29%5E2=2125%2F98+
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Answer by htmentor(1343) About Me  (Show Source):
You can put this solution on YOUR website!
The perpendicular bisectors of the sides of the triangle all intersect at a common point, the center of the circumscribed circle.
We need to first find the midpoints of two line segments joining the vertices.
midpoint AC = (3/2, -1)
midpoint BC = (3/2,5/2)
Now compute the slopes of the bisector lines
The slope of AC is (2 - -4)/(5 - -2) = 6/7
The slope of BC is (3 - 2)/(-2 - 5) = -1/7
Thus the equations for the bisector lines are
y + 1 = (-7/6)(x - 3/2) -> y = (-7/6)x + 3/4 [bisector of AC]
y - 5/2 = 7(x - 3/2) -> y = 7x - 8 [bisector of BC]
Now we just the point of intersection of these two lines to get the center of the circle:
(-7/6)x + 3/4 = 7x - 8
This gives x = 1.071
Thus y = 7*1.071 - 8 = -0.5
So the center is (1.071,-0.5)
The radius is the distance from the center to any of the vertices:
Using the distance formula gives R = sqrt(21.684)