A triangle has corners at #(4 ,6 )#, #(5 ,9 )#, and #(7 ,5 )#. What is the area of the triangle's circumscribed circle?

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Answer 1 · 2016-07-02T07:57:48

Area of circumscribed circle is #15.687#

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 #(4,6)#, #(5,9)# and #(7,5)#. This will be surely distance between pair of points, which is

#a=sqrt((5-4)^2+(9-6)^2)=sqrt(1+9)=sqrt10=3.1623#

#b=sqrt((7-5)^2+(5-9)^2)=sqrt(4+16)=sqrt20=4.4721# and

#c=sqrt((7-4)^2+(5-6)^2)=sqrt(9+1)=sqrt10=3.1623#

Hence #s=1/2(3.1623+4.4721+3.1623)=1/2xx10.7967=5.3984#

and #Delta=sqrt(5.3984xx(5.3984-3.1623)xx(5.3984-4.4721)xx(5.3984-3.1623)#

= #sqrt(5.3984xx2.2361xx0.9263xx2.2361)=sqrt25.0034=5.0034#

And radius of circumscribed circle is

#(3.1623xx4.4721xx3.1623)/(4xx5.0034)=2.2346#

And area of circumscribed circle is #3.1416xx(2.2346)^2=15.687#