A triangle has corners at #(6 ,8 )#, #(1 ,2 )#, and #(3 ,9 )#. What is the area of the triangle's circumscribed circle?

Question

A triangle has corners at #(6  ,8   )#, #(1   ,2   )#, and #(3   ,9  )#.  What is the area of the triangle's circumscribed circle?

Sophia Alfred · Accepted Answer

Area of the triangle's circumscribed circle is #48.005#.

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 #(6,8)#, #(1,2)# and #(3,9)#. This will be surely distance between pair of points, which is #a=sqrt((1-6)^2+(2-8)^2)=sqrt(25+36)=sqrt61=7.810# #b=sqrt((3-1)^2+(9-2)^2)=sqrt(4+49)=sqrt53=7.280# and #c=sqrt((3-6)^2+(9-8)^2)=sqrt(9+1)=sqrt10=3.162# Hence #s=1/2(7.810+7.280+3.162)=1/2xx18.252=9.126# and #Delta=sqrt(9.126xx(9.126-7.810)xx(9.126-7.280)xx(9.126-3.162)# = #sqrt(9.126xx1.316xx1.846xx5.964)=sqrt132.2226=11.499# And radius of circumscribed circle is #(7.810xx7.280xx3.162)/(4xx11.499)=3.909# And area of circumscribed circle is #3.1416xx(3.909)^2=48.005#

Allison Allen · Answer

undefined To find the area of the circumscribed circle of a triangle, you can use the formula:

Area = (abc) / (4R)

Where:
a, b, and c are the lengths of the sides of the triangle,
and R is the radius of the circumscribed circle.

First, you need to find the lengths of the sides of the triangle using the distance formula:

d = √((x2 - x1)^2 + (y2 - y1)^2)

Then, you can use Heron's formula to find the area of the triangle:

s = (a + b + c) / 2

Area = √(s * (s - a) * (s - b) * (s - c))

Once you have the area of the triangle, you can solve for R using the formula:

R = (abc) / (4 * Area)

Finally, you can calculate the area of the circumscribed circle using the formula for the area of a circle:

Area = π * R^2.