Question

A triangle has corners at `(6 ,4 )`, `(8 ,2 )`, and `(3 ,1 )`. What is the area of the triangle's circumscribed circle?

Answer 1

Area of circumscribed circle is `20.42` sq.unit.

Explanation:

The three corners are `A (6,4) B (8,2) and C (3,1)`

Distance between two points `(x_1,y_1) and (x_2,y_2)` is

`D= sqrt ((x_1-x_2)^2+(y_1-y_2)^2`

Side `AB= sqrt ((6-8)^2+(4-2)^2)=sqrt(8) ~~ 2.83`unit

Side `BC= sqrt ((8-3)^2+(2-1)^2)=sqrt(26) ~~5.10`unit

Side `CA= sqrt ((3-6)^2+(1-4)^2)=sqrt(18) ~~ 4.24`unit

Area of Triangle is `A_t = |1/2(x1(y2−y3)+x2(y3−y1)+x3(y1−y2))|`

`A_t = |1/2(6(2−1)+8(1−4)+3(4−2))|` or

`A_t = |1/2(6-24+6)| = |-6| =6.0` sq.unit.

Radius of circumscribed circle is `R=(AB*BC*CA)/(4*A_t)` or

`R=(sqrt(8)*sqrt(26)*sqrt(18))/(4*6) ~~ 2.55` unit

Area of circumscribed circle is `A_c=pi*R^2=pi*2.55^2~~20.42`

sq.unit [Ans]