A triangle has corners at #(4 ,6 )#, #(2 ,9 )#, and #(8 ,4 )#. What is the area of the triangle's circumscribed circle?

Question

A triangle has corners at #(4  ,6   )#, #(2   ,9   )#, and #(8   ,4  )#.  What is the area of the triangle's circumscribed circle?

Noah Atkinson · Accepted Answer

Area of circumscribed circle is #194.5068#

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)#, #(2,9)# and #(8,4)#. This will be surely distance between pair of points, which is

#a=sqrt((2-4)^2+(9-6)^2)=sqrt(4+9)=sqrt13=3.6056#

#b=sqrt((8-2)^2+(4-9)^2)=sqrt(36+25)=sqrt61=7.8102# and

#c=sqrt((8-4)^2+(4-6)^2)=sqrt(16+4)=sqrt20=4.4721#

Hence #s=1/2(3.6056+7.8102+4.4721)=1/2xx15.8879=7.944#

and #Delta=sqrt(7.944xx(7.944-3.6056)xx(7.944-7.8102)xx(7.944-4.4721)#

= #sqrt(7.944xx4.3384xx0.1338xx3.4719)=sqrt16.01=4.0013#

And radius of circumscribed circle is

#(3.6056xx7.8102xx4.4721)/(4xx4.0013)=7.8685#

And area of circumscribed circle is #3.1416xx(7.8685)^2=194.5068#

Noah Atkinson · Answer

undefined To find the area of the circumscribed circle of a triangle, we first need to find the lengths of the sides of the triangle. Then, we can use these side lengths to calculate the radius of the circumscribed circle. Once we have the radius, we can use the formula for the area of a circle to find the area of the circumscribed circle.

Using the given coordinates, we can calculate the lengths of the sides of the triangle using the distance formula:

Side 1: $ \sqrt{(4 - 2)^2 + (6 - 9)^2} = \sqrt{4 + 9} = \sqrt{13} $
Side 2: $ \sqrt{(4 - 8)^2 + (6 - 4)^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5} $
Side 3: $ \sqrt{(2 - 8)^2 + (9 - 4)^2} = \sqrt{36 + 25} = \sqrt{61} $

Next, we can use these side lengths to find the radius of the circumscribed circle using the formula:

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

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

Using Heron's formula to find the area of the triangle:

$ s = \frac{a + b + c}{2} = \frac{\sqrt{13} + 2\sqrt{5} + \sqrt{61}}{2} $

$ A = \sqrt{s(s - a)(s - b)(s - c)} $

Finally, once we have the radius, we can use the formula for the area of a circle to find the area of the circumscribed circle:

$ Area = \pi R^2 $