### Instruction: What is the radius of the circle inscribed in triangle $ABC$ if $AB = 12, AC=14, BC=16$? Express your answer in simplest radical form.### Response: I know that the radius of the inscribed circle is related to the area and perimeter of the triangle, by the formula $r = \frac{A}{s}$, where $A$ is the area and $s$ is the semiperimeter.
To find the area of the triangle, I can use Heron's formula, which says that $A = \sqrt{s(s-a)(s-b)(s-c)}$, where $a, b, c$ are the side lengths.
In this case, $s = \frac{12+14+16}{2} = 21$, so $A = \sqrt{21(21-12)(21-14)(21-16)} = \sqrt{21(9)(7)(5)} = \sqrt{6615}$.
Therefore, $r = \frac{\sqrt{6615}}{21}$.
To simplify the radical, I can look for any perfect square factors in the radicand.
I notice that $6615 = 5 \cdot 1323 = 5 \cdot 3 \cdot 441 = 5 \cdot 3 \cdot 21 \cdot 21$, so $\sqrt{6615} = \sqrt{5 \cdot 3 \cdot 21^2} = 21 \sqrt{15}$.
Hence, $r = \frac{21 \sqrt{15}}{21} = \sqrt{15}$.