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 $3.1416xx(2.2346)^2=15.687$

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