A triangle has corners at #(8 ,7 )#, #(2 ,1 )#, and #(5 ,6 )#. What is the area of the triangle's circumscribed circle?

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Answer 1 · 2017-07-12T11:46:55

The area of the circle is #=133.5u^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,

#(8-a)^2+(7-b)^2=r^2#.......#(1)#

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

#(5-a)^2+(6-b)^2=r^2#.........#(3)#

We have #3# equations with #3# unknowns

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

#64-16a+a^2+49-14b+b^2=4-4a+a^2+1-2b+b^2#

#12a+12b=108#

#a+b=9#.............#(4)#

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

#4-4a+a^2+1-2b+b^2=25-10a+a^2+36-12b+b^2#

#6a+10b=56#

#3a+5b=28#..............#(5)#

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

#3(9-b)+5b=28#

#27-3b+5b=28#

#2b=1#, #=>#, #b=1/2#

#a=9-1/2#, #=>#, #a=17/2#

The center of the circle is #=(17/2,1/2)#

#r^2=(2-17/2)^2+(1-1/2)^2=(-13/2)^2+(1/2)^2#

#=170/4#

#=85/2#

The area of the circle is

#A=pi*r^2=pi*85/2=133.5#