Question

A triangle has corners at `(9 ,8 )`, `(2 ,3 )`, and `(7 ,4 )`. What is the area of the triangle's circumscribed circle?

Answer 1

`"Area" = 2405/81pi`

Explanation:

The standard Cartesian form for the equation of a circle is:

`(x-h)^2+(y-k)^2=r^2`

Where `(h,k)` is the center point and `r` is the radius.

We can use the equation and the 3 points to write 3 equations:

`(9-h)^2+(8-k)^2=r^2" [1]"`
`(2-h)^2+(3-k)^2=r^2" [2]"`
`(7-h)^2+(4-k)^2=r^2" [3]"`

Expand the squares:

`81-18h+h^2+64-16k+k^2=r^2" [1.1]"`
`4-4h+h^2+9-6k+k^2=r^2" [2.1]"`
`49-14h+h^2+16-8k+k^2=r^2" [3.1]"`

Subtract equation [2.1] from equation [1.1]:

`-14h-10k+ 132 = 0`

Write in standard form:

`7h + 5k = 66" [4]"`

Subtract equation [2.1] from equation [3.1]:

`-10h -2k + 52 = 0`

Write as k in terms of h:

`-2k =-52 +10h`

`k = 26 -5h" [5]"`

Substitute equation [5] into equation [4]:

`7h + 5(26 -5h) = 66`

`7h + 130 - 25h = 66`

`-18h=-64`

`h = 32/9`

Use equation [5] to find the value of k:

`k = 26 -5(32/9)`

`k = 74/9`

Use equation [1] to solve for `r^2`

`(9-32/9)^2+(8-74/9)^2=r^2`

`r^2 = 2405/81`

Because the area of a circle is:

`"Area" = pir^2`

We need only multiply by `pi`

`"Area" = 2405/81pi`