A triangle has corners at #(7 ,3 )#, #(2 ,5 )#, and #(1 ,4 )#. What is the area of the triangle's circumscribed circle?

1 Answer

Area #A=1073/98 *pi#=34.39723385" "#square units

Explanation:

Let #A(1, 4)# , #B(7, 3)#, #C(2, 5)#

One solution is to determine the orthocenter #H(x_H, y_H)#, then solve for the radius R

#R=sqrt((x_A-x_H)^2+(y_A-y_H)^2)#

the circumcenter is the intersection of the line segment bisectors of the side AB and side AC which have the midpoints at #(4, 7/2)# and #(3/2, 9/2)# respectively.

the equation of the line segment bisector at AB is

#y-7/2=-1/m_(AB)*(x-4)" "#where #" "m_(AB) = -1/6#

#y-7/2=6*(x-4)#

simplified
#12x-2y=41#

the equation of the line segment bisector at AC is

#y-9/2=-1/m_(AC)*(x-3/2)" "#where #m_(AC)=1#

#y-9/2=-1*(x-3/2)#

simplified

#x+y=6#

simultaneous solution using
#x+y=6# and #12x-2y=41# yields

#H(x_H, y_H)=(53/14, 31/14)#

Solve for R now, using any vertex of the triangle ...
let us choose A(1, 4)

#R=sqrt((1-53/14)^2+(4-31/14)^2)#

#R=sqrt(2146)/14#

it follows ,....the area of the circle with center (53/14, 31/14) radius R (see the orange line segment)

Desmos.com

#A=pi*R^2#
#A=pi*(sqrt(2146)/14)^2#
#A=1073/98 *pi#=34.39723385" "#square units

God bless....I hope the explanation is useful.