A triangle has corners at #(7 ,3 )#, #(5 ,8 )#, and #(4 ,6 )#. What is the area of the triangle's circumscribed circle?

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Answer 1 · 2016-05-24T10:10:12

Area of triangle's circumscribed circle is #25.16#

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 #(7,3)#, #(5,8)# and #(4,6)#. This will be surely distance between pair of points, which is

#a=sqrt((7-5)^2+(3-8)^2)=sqrt(4+25)=sqrt29=5.385#

#b=sqrt((4-5)^2+(6-8)^2)=sqrt(1+4)=sqrt5=2.236# and

#c=sqrt((7-4)^2+(3-6)^2)=sqrt(9+9)=sqrt18=4.243#

Hence #s=1/2(5.385+2.236+4.243)=1/2xx11.864=5.932#

and #Delta=sqrt(5.932xx(5.932-5.385)xx(5.932-2.236)xx(5.932-4.243)#

= #sqrt(5.932xx0.547xx3.696xx1.689)=sqrt20.3693=4.513#

And radius of circumscribed circle is

#(5.385xx2.236xx4.243)/(4xx4.513)=2.83#

And area of circumscribed circle is #3.1416xx(2.83)^2=25.16#