A triangle has corners at #(5 ,1 )#, #(3 ,9 )#, and #(4 ,7 )#. What is the area of the triangle's circumscribed circle?

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Answer 1 · 2017-06-12T18:24:00

The area of the circumscribed circle is #=855.3u^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,

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

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

#(4-a)^2+(7-b)^2=r^2#.........#(3)#

We have #3# equations with #3# unknowns

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

#25-10a+a^2+1-2b+b^2=9-6a+a^2+81-18b+b^2#

#4a-16b=-54#

#2a-8b=-27#.............#(4)#

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

#9-6a+a^2+81-18b+b^2=16-8a+a^2+49-14b+b^2#

#2a-4b=-25#

#2a-4b=-25#..............#(5)#

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

#-27+8b-4b=-25#

#4b=2#

#b=2/4=1/2#

#2a=-25+4b=-25+2=-23#, #=>#, #a=-23/2#

The center of the circle is #=(-23/2,1/2)#

#r^2=(5-a)^2+(1-b)^2=(5+23/2)^2+(1-1/2)^2#

#=1089/4+1/4#

#=1090/4#

The area of the circle is

#A=pi*r^2=1089/4*pi=855.3#