Question

A triangle has corners at `(3 ,1 )`, `(4 ,9 )`, and `(7 ,4 )`. What is the area of the triangle's circumscribed circle?

Answer 1

Area of circumscribed circle is `51.6` sq.unit.

Explanation:

The three corners are `A (3,1) B (4,9) and C (7,4)`

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 ((3-4)^2+(1-9)^2) ~~ 8.06`unit

Side `BC= sqrt ((4-7)^2+(9-4)^2) ~~5.83`unit

Side `CA= sqrt ((7-3)^2+(4-1)^2) = 5.0` unit

The semi perimeter of triangle is `s=(AB+BC+CA)/2` or

`s= (8.06+5.83+5.0)/2~~ 9.45` unit

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

`A_t = |1/2(3(9−4)+4(4−1)+7(1−9))|` or

`A_t = |1/2(15+12-56)| = | -29/2| =14.5` sq.unit

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

`R=(sqrt(65)*sqrt(34)*sqrt(25))/(4*14.5) ~~ 4.05`

Area of circumscribed circle is `A_c=pi*R^2=pi*4.05^2~~51.6`

sq.unit [Ans]

Answer 2

`pi r^2 = {pi (65)(34)(25) }/{ 4(25)(34) - (65-34-25)^2} = (27625 pi)/1682`

Explanation:

The other answer gives an approximation for this question that may be answered exactly. I don't blame the other answer; we've all been taught to do this. I prefer the exact answer, which generally means avoiding square roots.

The circumcircle is just the circle through the three vertices; the triangle almost doesn't matter. Except, miraculously, the circumradius `r` equals the product of the triangle sides `a,b,c` divided by four times the triangle's area `A`.

`r = {abc}/{4A}`

It's much more useful squared, and we're looking for `pi r^2` anyway.

`pi r^2 = {pi a^2 b^2 c^2}/{16A^2}`

The coordinates give the squared distances easily. Archimedes' Theorem relates the squared distances to the triangle area:

`16A^2 = 4a^2b^2 - (c^2-a^2-b^2)^2`

So,

`pi r^2 = {pi a^2 b^2 c^2}/{ 4a^2b^2 - (c^2-a^2-b^2)^2}`

We form the squared distances from pairs of points `(3,1),(4,9),(7,4)`

`c^2=(4-3)^2+(9-1)^2=65`

`a^2=(7-4)^2+(4-9)^2=34`

`b^2=(7-3)^2+(4-1)^2=25`

We're free to play with the assignment to `a,b` and `c.` I prefer making `c` the biggest one which gives me the smallest thing to square. (I try to do these by hand sometimes to keep in shape.)

`pi r^2 = {pi (65)(34)(25) }/{ 4(25)(34) - (65-34-25)^2} = (27625 pi)/1682`

The other answer said `51.6,` this is `51.597203956847822956320... quad sqrt`