# A triangle has corners at (8 ,7 ), (2 ,1 ), and (3 ,6 ). What is the area of the triangle's circumscribed circle?

Jun 21, 2018

$A \approx 5.15$

#### Explanation:

First we need to find the length of each side, to do this we need to use the distance formula on pairs of ordered pairs:

$d = \sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2}}$

$d = \sqrt{{\left(8 - 2\right)}^{2} + {\left(7 - 1\right)}^{2}} = 6 \sqrt{2}$

$d = \sqrt{{\left(2 - 3\right)}^{2} + {\left(1 - 6\right)}^{2}} = \sqrt{26}$

$d = \sqrt{{\left(3 - 8\right)}^{2} + {\left(6 - 7\right)}^{2}} = \sqrt{26}$

Now use the formula for a triangle inscribed circle:

$s = \frac{a + b + c}{2}$

$r = \sqrt{\frac{\left(s - a\right) \left(s - b\right) \left(s - c\right)}{s}}$

Plug in our values:

$s = \frac{6 \sqrt{2} + \sqrt{26} + \sqrt{26}}{2}$

$s \approx 9.34$

$r = \sqrt{\frac{\left(9.34 - 6 \sqrt{2}\right) \left(9.34 - \sqrt{26}\right) \left(9.34 - \sqrt{26}\right)}{9.34}}$

$r \approx 1.28$

$A = \pi {r}^{2}$

$A = \pi {\left(1.28\right)}^{2}$

$A \approx 5.15$