A triangle has corners at #(6 ,8 )#, #(1 ,2 )#, and #(3 ,9 )#. What is the area of the triangle's circumscribed circle?

1 Answer
May 27, 2016

Area of the triangle's circumscribed circle is #48.005#.

Explanation:

If the sides of a triangle are #a#, #b# and #c#, then the area of the triangle #Delta# is given by the formula

#Delta=sqrt(s(s-a)(s-b)(s-c))#, where #s=1/2(a+b+c)#

and radius of circumscribed circle is #(abc)/(4Delta)#

Hence let us find the sides of triangle formed by #(6,8)#, #(1,2)# and #(3,9)#. This will be surely distance between pair of points, which is

#a=sqrt((1-6)^2+(2-8)^2)=sqrt(25+36)=sqrt61=7.810#

#b=sqrt((3-1)^2+(9-2)^2)=sqrt(4+49)=sqrt53=7.280# and

#c=sqrt((3-6)^2+(9-8)^2)=sqrt(9+1)=sqrt10=3.162#

Hence #s=1/2(7.810+7.280+3.162)=1/2xx18.252=9.126#

and #Delta=sqrt(9.126xx(9.126-7.810)xx(9.126-7.280)xx(9.126-3.162)#

= #sqrt(9.126xx1.316xx1.846xx5.964)=sqrt132.2226=11.499#

And radius of circumscribed circle is

#(7.810xx7.280xx3.162)/(4xx11.499)=3.909#

And area of circumscribed circle is #3.1416xx(3.909)^2=48.005#