problem: What is the circumference of the circle with center at $P$ and passing through $Q$? Express your answer in terms of $\pi$.
  
  [asy]
  size(150); pair P = (-2,3), Q=(10,-2); string stringpair(pair p){return "$("+string(p.x)+", "+string(p.y)+"$)";}
  draw((-15,0)--(15,0),Arrows(4)); draw((0,-15)--(0,15),Arrows(4));
  dot("$Q$"+stringpair(Q),Q,SE,linewidth(3)); dot("$P$"+stringpair(P),P,NW,linewidth(3));
  
  [/asy]
solution: To find the circumference, we must first find the length of the radius, $PQ$. We can do this by using the distance formula or by noticing that the points $P$, $Q$, and $(-2, -2)$ form a right triangle with legs of length 5 and 12. This is a Pythagorean triple, so the hypotenuse $PQ$ must equal 13.
  
  Now that we know the radius has length 13, the circumference is $2 \cdot 13 \pi = \boxed{26\pi}$.