A triangle has corners at (8,3), (2,4), and (7,2). What is the area of the triangle's circumscribed circle?

1 Answer
Dean R.
Apr 18, 2018

The circle circumscribed on a triangle is the one that passes through the three vertices.

The general equation for a circle with center (a,b) and squared radius k is

(x−a)2+(y−b)2=k

Substituting our three points,

(8−a)2+(3−b)2=k

(2−a)2+(4−b)2=k

(7−a)2+(2−b)2=k

a2+b2−16a−6b+64+9=k

a2+b2−4a−8b+4+16=k

a2+b2−14a−4b+49+4=k

Subtracting pairs,

−12a+2b+53=0

−2a−2b+20=0

Adding,

−14a+73=0

a=7314

12a+12b−120=0

14b−67=0

b=6714

k=(2−7314)2+(4−6714)2

k=1142((28−73)2+(56−67)2)=2146142=107398

Usually they pick nicer numbers for these problems. Our circle's area is A=πr2=πk

A=1073π98

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