### Instruction: What is the radius of the circle inscribed in triangle $ABC$ if $AB = AC=7$ and $BC=6$? Express your answer in simplest radical form.### Response: To find the radius of the inscribed circle, I need to use the formula $r = A/s$, where $r$ is the radius, $A$ is the area of the triangle, and $s$ is the semiperimeter of the triangle.
To apply this formula, I need to find the area and the semiperimeter of the triangle.
The semiperimeter is easy: it is just half of the perimeter, which is $7+7+6=20$.
So $s=20/2=10$.
To find the area, I need to use Heron's formula, since I don't have a right angle or the height of the triangle.
Heron's formula says that the area of a triangle with sides $a,b,c$ and semiperimeter $s$ is $\sqrt{s(s-a)(s-b)(s-c)}$.
In this case, $a=b=7$ and $c=6$.
So the area is $\sqrt{10(10-7)(10-7)(10-6)}$.
Simplifying, I get $\sqrt{10(3)(3)(4)}=\sqrt{360}=6\sqrt{10}$.
Now I have the area and the semiperimeter, so I can plug them into the formula for the radius: $r = A/s = 6\sqrt{10}/10$.