A triangle has corners at #(5 ,2 )#, #(8 ,1 )#, and #(3 ,4 )#. What is the area of the triangle's circumscribed circle?

1 Answer
ali ergin
Dec 7, 2016

#"area is 42.5"pi#

Explanation:

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#"the points A,B,C are on the circle."#
#"we can write the fallowing equations"#

#A(5,2)#

#x^2+y^2+ax+bx+c=0#
#5^2+2^2+5a+2b+c=0#
#25+4+5a+2b+c=0#

#5a+2b+c=-29" (1)"#

#B(8,1)#

#x^2+y^2+ax+bx+c=0#
#8^2+1^2+8a+b+c=0#
#64+1+8a+b+c=0#

#8a+b+c=-65" (2)"#

#C(3,4)#

#x^2+y^2+ax+bx+c=0#
#3^2+4^2+3a+4b+c=0#
#9+16+3a+4b+c=0#

#3a+4b+c=-25" (3)"#

#"let's add the equation (2) to (3)"#
#11a+5b+2c=-90" (4)"#

#"expand the equation (1) by 2"#
#10a+4b+2c=-58" (5)"#

#"subtract the equation (5) from (4)"#

#a+b=-32" (5)"#

#"subtract the equation (1) from (2)"#

#3a-b=-36" (6)"#

#"now add (5) to (6)"#

#4a=-68#

#color(red)(a=-17)#

#"use (5)"#

#-17+b=-32#

#b=-32+17#

#color(red)(b=-15)#

#"use (2)"#

#8(-17)-15+c=-65#

#-136-15+c=-65#

#c=-65+151#

#color(red)(c=86)#

#"r:radius of circle"#

#r=sqrt(a^2+b^2-4c)/2#

#r=sqrt(289+225-344)/2#

#r=sqrt(514-344)#

#r=sqrt(170)/2#

#r^2=170/4=42.5#

#area=pi*r^2#

#area=42.5pi#

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