A triangle has corners at #(1 ,5 )#, #(7 ,9 )#, and #(4 ,2 )#. What is the area of the triangle's circumscribed circle?

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Answer 1 · 2017-07-27T07:32:46

The area of the circumscribed circle is #=47.38u^2#

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,

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

#(7-a)^2+(9-b)^2=r^2#..........#(2)#

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

We have #3# equations with #3# unknowns

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

#1-2a+a^2+25-10b+b^2=49-14a+a^2+81-18b+b^2#

#12a+8b=130-26=104#

#3a+2b=26#.............#(4)#

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

#16-8a+a^2+4-4b+b^2=49-14a+a^2+81-18b+b^2#

#6a+14b=130-20=110#

#3a+7b=55#..............#(5)#

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

#26-2b=55-7b#, #=>#, #5b=29#, #b=29/5#

#3a=26-2b=26-58/5=72/5#, #=>#, #a=24/5#

The center of the circle is #=(24/5,29/5)#

#r^2=(1-24/5)^2+(5-29/5)^2=(-19/5)^2+(-4/5)^2#

#=361/25+16/25#

#=377/25#

The area of the circle is

#A=pi*r^2=pi*377/25=47.38u^2#