# A triangle has corners at (3 , 5 ), (4 ,7 ), and (8 ,6 ). What is the radius of the triangle's inscribed circle?

Oct 9, 2017

Coordinates of incenter are (4.42, 6.09)

#### Explanation:

Let BC = a, AB = c, CA = b & Perimeter = p;
Let the incenter point be O.
$a = \sqrt{{\left(3 - 4\right)}^{2} + {\left(5 - 7\right)}^{2}} = \sqrt{5} = 2.236$
$b = \sqrt{{\left(8 - 4\right)}^{2} + {\left(6 - 7\right)}^{2}} = \sqrt{17} = 4.123$
$c = \sqrt{{\left(8 - 3\right)}^{2} + {\left(6 - 5\right)}^{2}} = \sqrt{26} = 5.099$

$p = \sqrt{5} + \sqrt{17} + \sqrt{26} = 2.236 + 4.123 + 5.099 = 11.458$

$O x = \frac{a A x + b B x + c C x}{p}$
$O x = \frac{\left(2.236 \cdot 8\right) + \left(4.123 \cdot 3\right) + \left(5.099 \cdot 4\right)}{11.458}$
$O x = \frac{17.888 + 12.369 + 20.396}{11.458} =$ 4.421

$O y = \frac{a A y + b B y + c C y}{p}$
$O y = \frac{\left(2.236 \cdot 6\right) + \left(4.123 \cdot 5\right) + \left(5.099 \cdot 7\right)}{11.458}$
$O y = \frac{13.416 + 20.615 + 35.693}{11.458} =$ 6.085