A triangle has corners at (9,8), (2,3), and (1,4). What is the area of the triangle's circumscribed circle?

1 Answer
Narad T.
Jun 6, 2017

The area is =64.6

Explanation:

To calculate the area of the circle, we must calculate the radius r of the circle

Let the center of the circle be O=(a,b)

Then,

(9−a)2+(8−b)2=r2.......(1)

(2−a)2+(3−b)2=r2..........(2)

(1−a)2+(4−b)2=r2.........(3)

We have 3 equations with 3 unknowns

From (1) and (2), we get

81−18a+a2+64−16b+b2=4−4a+a2+9−6b+b2

14a+10b=132

7a+5b=66.............(4)

From (2) and (3), we get

4−4a+a2+9−6b+b2=1−2a+a2+16−8b+b2

2a−2b=−4

a−b=−2..............(5)

From equations (4) and (5), we get

7(b−2)+5b=66

12b=80

b=8012=203

a=b−2=203−2=143

The center of the circle is =(143,203)

r2=(1−a)2+(4−b)2=(1−143)2+(4−203)2

=11232+8232

=20.56

The area of the circle is

A=π⋅r2=20.56⋅π=64.6

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