Question

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

Answer 1

`(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`