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

1 Answer
bp
Jan 14, 2017

(65pi)/265π2

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)]12[x1(y2−y3)+x2(y3−y1)+x3(y1−y2)]
=1/2[6(6-6)+7(6-4)+3(4-6]= 812[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√(6−7)2+(4−6)2=√5,
sqrt((7-3)^2 +(6-6)^2)= sqrt16√(7−3)2+(6−6)2=√16
and sqrt((6-3)^2 +(4-6)^2)= sqrt 13√(6−3)2+(4−6)2=√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

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