### Instruction: What is the radius of the circle inscribed in triangle $ABC$ if $AB = 5, AC=6, BC=7$? Express your answer in simplest radical form.### Response: I remember that the radius of the circle inscribed in a triangle is the area of the triangle divided by its semiperimeter.
That's right. Let's first find the area of the triangle.
We can use Heron's formula. The semiperimeter is $(5+6+7)/2=9$. Therefore, the area is $\sqrt{9(9-5)(9-6)(9-7)}=\sqrt{9\cdot4\cdot3\cdot2}=6\sqrt{6}$.
Now we can divide this by the semiperimeter to find the radius.