A triangle has corners at #(9 ,7 )#, #(2 ,1 )#, and #(5 ,2 )#. What is the area of the triangle's circumscribed circle?

• The corner coordinates of the ABC triangle are on the circumference circle.
• the first step is to find the edge lengths of triangle a, b, c.
• We can find the distance between two known coordinates by using the following formula.

#P_1(x_1,y_1)" , "P_2(x_2,y_2)#

#l=sqrt((x_2-x_1)^2+(y_2-y_1)^2)#

1. The length of a side:
   #a=sqrt((5-2)^2+(2-1)^2)=sqrt(3^2+1^2)=sqrt(9+1)=sqrt(10)" units"#

2. The length of b side:
   #b=sqrt((9-5)^2+(7-2)^2)=sqrt(4^2+5^2)=sqrt(16+25)=sqrt(41)" units"#

3. The length of c side:
   #c=sqrt((9-2)^2+(7-1)^2)=sqrt(7^2+6^2)=sqrt(49+36)=sqrt(85)" units"#

-In the second step, we can calculate the area of the triangle known as corner coordinates.

#A(x_1,y_1)" , "B(x_2,y_2)" , "C=(x_3,y_3)#
#A(9,7)" , "B(2,1)" , "C(5,2)#

#"triangle's area="1/2*|9*1+2*2+5*7-2*7-5*1-9*2 |#

#"triangle's area="1/2*|9+4+35-14-5-18 |#

#"triangle's area="1/2*|9+4+35-14-5-18 |=5.5" units"^2#

• now we can use the formula given below.

#area(ABC)=(a*b*c)/(4*r)#

#5.5=(sqrt (10)*sqrt(41)*sqrt(85))/(4*r)#

#22r=sqrt(10*41*85)#

#r=(sqrt(10*41*85))/22#

#r=(sqrt(34850))/22#

#r=8.49" units"#

• the area of the triangle's circumscribed circle:

#area=pi*r^2#

#area=3.14*(8.49)^2#

#area=226.33" units"^2#