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

1 Answer
Jan 14, 2017

#(65pi)/2#

Explanation:

First, use the formula for area of triangle=#1/2 [x_1 (y_2 -y_3)+x_2 (y_3 -y_1) +x_3 (y_1 -y_2)]#
=#1/2[6(6-6)+7(6-4)+3(4-6]= 8#

Next find the side lengths using distance formula. The sides would be #sqrt( (6-7)^2 +(4-6)^2)= sqrt5#,
#sqrt((7-3)^2 +(6-6)^2)= sqrt16#
and #sqrt((6-3)^2 +(4-6)^2)= sqrt 13#

Now use the formula #R= (abc)/(4Delta)# to get the radius R of the circumcircle of the triangle, where a,b,c aree its sides and #Delta# is the area of the triangle.

Accordingly, #R=(sqrt5 sqrt16 sqrt13)/(4(8))#

Area of the circumcircle would thus be #pi R^2#

= #(pi (5)(13)(16))/32#= #(65pi)/2#