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

Question

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

Maverick Smith · Accepted Answer

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#

Autumn Armstrong · Answer

undefined To find the area of the circumscribed circle of a triangle, you need to determine the radius of the circumscribed circle first. Then, you can use the formula for the area of a circle, which is π times the square of the radius. To find the radius, you can use the distances between the vertices of the triangle and the circumcenter, which is the center of the circumscribed circle. Once you have the radius, you can calculate the area of the circle using the formula mentioned earlier.

Sophia Alfred · Answer

undefined To find the area of the circumscribed circle of a triangle given its vertices, we can use the formula for the circumradius ($R$) of a triangle:

$$ R = \frac{abc}{4A} $$

Where $a$, $b$, and $c$ are the side lengths of the triangle, and $A$ is the area of the triangle.

We first need to find the side lengths of the triangle using the given vertices, and then calculate the area using Heron's formula. Once we have the side lengths and the area, we can find the circumradius.

Given the vertices:
$ A(4, 6) $
$ B(5, 9) $
$ C(7, 5) $

We can find the side lengths:
$ AB = \sqrt{(5-4)^2 + (9-6)^2} $
$ BC = \sqrt{(7-5)^2 + (5-9)^2} $
$ CA = \sqrt{(4-7)^2 + (6-5)^2} $

Then, using Heron's formula, the area of the triangle $ABC$ is:

$$ A = \sqrt{s(s-AB)(s-BC)(s-CA)} $$

Where $ s $ is the semiperimeter, given by $ s = \frac{AB + BC + CA}{2} $.

Once we have the area, we can find the circumradius using the formula $ R = \frac{abc}{4A} $, where $ a $, $ b $, and $ c $ are the side lengths of the triangle.

After finding the circumradius, the area of the circumscribed circle is given by $ \pi R^2 $.