# A triangle has sides with lengths of 8, 7, and 6. What is the radius of the triangles inscribed circle?

Jan 25, 2016

If $a , b \mathmr{and} c$ are the three sides of a triangle then the radius of its in center is given by

$R = \frac{\Delta}{s}$

Where $R$ is the radius $\Delta$ is the are of the triangle and $s$ is the semi perimeter of the triangle.

The area $\Delta$ of a triangle is given by

Delta=sqrt(s(s-a)(s-b)(s-c)

And the semi perimeter $s$ of a triangle is given by
$s = \frac{a + b + c}{2}$

Here let $a = 8 , b = 7 \mathmr{and} c = 6$

$\implies s = \frac{8 + 7 + 6}{2} = \frac{21}{2} = 10.5$

$\implies s = 10.5$

$\implies s - a = 10.5 - 8 = 2.5 , s - b = 10.5 - 7 = 3.5 \mathmr{and} s - c = 10.5 - 6 = 4.5$

$\implies s - a = 2.5 , s - b = 3.5 \mathmr{and} s - c = 4.5$

$\implies \Delta = \sqrt{10.5 \cdot 2.5 \cdot 3.5 \cdot 4.5} = \sqrt{413.4375} = 20.333$

$\implies R = \frac{20.333}{10.5} = 1.9364$ units

Hence, the radius of inscribed circle of the triangle is $1.9364$ units long.