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

1 Answer
Sunayan S
Oct 2, 2017

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)

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