Question

A triangle has corners at `(2 ,4 )`, `(6 ,5 )`, and `(3 ,3 )`. What is the area of the triangle's circumscribed circle?

Answer 1

See the answer below...

Explanation:

Suppose the circumcentre of the triangle is `(x,y)`

$1st STEP$
Hence the distance of the point of the circumcentre from the corners of the triangle will be same...
So we can write ,
`sqrt((x-2)^2+(y-4)^2)=sqrt((x-6)^2+(y-5)^2)=sqrt((x-3)^2+(y-3)^2)`
[Squaring each part and removing `x^2` and `y^2` from each part]

`=>-4x+4-8y+16=-12x+36-10y+25=-6x+9-6y+9`
Hence we get equations ...
`i)` `8x+2y=41`[From 1st and 2nd equation ]
`ii) 6x+4y=43`[From 2nd and 3rd equation]

From these equations we get
`16x+4y-6x-4y=82-43`
`=>x=3.9`
Similarly we get the value of `y=(41-8xx3.9)/2=4.9`

Thus we can determine the value of the radius of the circle...
`sqrt((3.9-2)^2+(4.9-2)^2)=2.1 unit` (approx)
Hence the area of the circle is `pixxr^2=13.86 unit^2` (approx)