Question

A triangle has corners at `(5 ,2 )`, `(8 ,1 )`, and `(3 ,4 )`. What is the area of the triangle's circumscribed circle?

Answer 1

`"area is 42.5"pi`

Explanation:

`"the points A,B,C are on the circle."`
`"we can write the fallowing equations"`

`A(5,2)`

`x^2+y^2+ax+bx+c=0`
`5^2+2^2+5a+2b+c=0`
`25+4+5a+2b+c=0`

`5a+2b+c=-29" (1)"`

`B(8,1)`

`x^2+y^2+ax+bx+c=0`
`8^2+1^2+8a+b+c=0`
`64+1+8a+b+c=0`

`8a+b+c=-65" (2)"`

`C(3,4)`

`x^2+y^2+ax+bx+c=0`
`3^2+4^2+3a+4b+c=0`
`9+16+3a+4b+c=0`

`3a+4b+c=-25" (3)"`

`"let's add the equation (2) to (3)"`
`11a+5b+2c=-90" (4)"`

`"expand the equation (1) by 2"`
`10a+4b+2c=-58" (5)"`

`"subtract the equation (5) from (4)"`

`a+b=-32" (5)"`

`"subtract the equation (1) from (2)"`

`3a-b=-36" (6)"`

`"now add (5) to (6)"`

`4a=-68`

`color(red)(a=-17)`

`"use (5)"`

`-17+b=-32`

`b=-32+17`

`color(red)(b=-15)`

`"use (2)"`

`8(-17)-15+c=-65`

`-136-15+c=-65`

`c=-65+151`

`color(red)(c=86)`

`"r:radius of circle"`

`r=sqrt(a^2+b^2-4c)/2`

`r=sqrt(289+225-344)/2`

`r=sqrt(514-344)`

`r=sqrt(170)/2`

`r^2=170/4=42.5`

`area=pi*r^2`

`area=42.5pi`