Question

A triangle has corners at `(5 ,1 )`, `(3 ,9 )`, and `(4 ,7 )`. What is the area of the triangle's circumscribed circle?

Answer 1

The area of the circumscribed circle is `=855.3u^2`

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,

`(5-a)^2+(1-b)^2=r^2`.......`(1)`

`(3-a)^2+(9-b)^2=r^2`..........`(2)`

`(4-a)^2+(7-b)^2=r^2`.........`(3)`

We have `3` equations with `3` unknowns

From `(1)` and `(2)`, we get

`25-10a+a^2+1-2b+b^2=9-6a+a^2+81-18b+b^2`

`4a-16b=-54`

`2a-8b=-27`.............`(4)`

From `(2)` and `(3)`, we get

`9-6a+a^2+81-18b+b^2=16-8a+a^2+49-14b+b^2`

`2a-4b=-25`

`2a-4b=-25`..............`(5)`

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

`-27+8b-4b=-25`

`4b=2`

`b=2/4=1/2`

`2a=-25+4b=-25+2=-23`, `=>`, `a=-23/2`

The center of the circle is `=(-23/2,1/2)`

`r^2=(5-a)^2+(1-b)^2=(5+23/2)^2+(1-1/2)^2`

`=1089/4+1/4`

`=1090/4`

The area of the circle is

`A=pi*r^2=1089/4*pi=855.3`