AD , BD, and CD are angle bisectors of the vertex angles of △ ABC. AG=5 meters and AD=13 meters.What is DF?Enter your answer in the box.

12    
This problem can be either trivial, or extremely complicated, depending upon if you were paying attention in class. The intersection of the angle bisectors of a triangle is called the incenter of the triangle. And the incenter of a triangle is where you can place the center of the largest circle that will be tangent to all three sides of the triangle. Because of that, the line segments DF, DG, and DE will all have the same length. Now look closely at triangle ADG, you'll notice that it's a right triangle, the leg AG has a length of 5 meters and the hypotenuse has a length of 13 meters. So using the Pythagorean theorem, we can easily get the length of DG and therefore DF. So:  
sqrt(13^2 - 5^2) = sqrt(169 - 25) = sqrt(144) = 12    
So the length of DF is 12.