A triangle has corners at #(4 ,7 )#, #(3 ,4 )#, and #(6 ,9 )#. What is the area of the triangle's circumscribed circle?

Question

A triangle has corners at #(4  ,7   )#, #(3   ,4   )#, and #(6   ,9  )#.  What is the area of the triangle's circumscribed circle?

Jayden Jackson · Accepted Answer

Area of circumscribed circle is #133.47#

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 #(4,7)#, #(3,4)# and #(6,9)#. This will be surely distance between pair of points, which is

#a=sqrt((3-4)^2+(4-7)^2)=sqrt(1+9)=sqrt10=3.1623#

#b=sqrt((6-3)^2+(9-4)^2)=sqrt(9+25)=sqrt34=5.831# and

#c=sqrt((6-4)^2+(9-7)^2)=sqrt(4+4)=sqrt8=2.8284#

Hence #s=1/2(3.1623+5.831+2.8284)=1/2xx11.8217=5.9109#

and #Delta=sqrt(5.9109xx(5.9109-3.1623)xx(5.9109-5.831)xx(5.9109-2.8284))#

= #sqrt(5.9109xx2.7486xx0.0799xx3.0825)=sqrt4.0014=2.0004#

And radius of circumscribed circle is

#(3.1623xx5.831xx2.8284)/(4xx2.0004)=6.518#

And area of circumscribed circle is #3.1416xx(6.518)^2=133.47#

Savannah Aguilar · Answer

undefined To find the area of the triangle's circumscribed circle, you first need to find the circumradius (the radius of the circle circumscribing the triangle). Then, you can use the formula for the area of a circle, which is $A = \pi r^2$, where $r$ is the radius of the circle.

The circumradius can be found using the formula:

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

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

First, find the side lengths of the triangle using the given coordinates and the distance formula. Then, use Heron's formula to find the area of the triangle. Finally, substitute the side lengths and area into the circumradius formula to find $R$.

Once you have $R$, you can use the formula $A = \pi R^2$ to find the area of the circumscribed circle.

This process will give you the area of the circumscribed circle of the triangle.