[ $\quad \underline{\text { U }}$ median duplication ]

The sides of a triangle are 11, 13, and 12. Find the median drawn to the largest side.

#
Use the theorem about the sum of the squares of the diagonals of a parallelogram or the formula for the median of a triangle.

## Solution

Let $A M$ be the median of the right triangle $A B C$, where $A B=12, A C=11, B C=13$. On the extension of the median $A M$ beyond point $M$, lay off the segment $M K$ equal to $A M$. Then $A B K C$ is a parallelogram. By the theorem on the sum of the squares of the diagonals of a parallelogram,

$$
A K^{2}+B C^{2}=2 A B^{2}+2 A C^{2}
$$

from which

$$
A K^{2}=2 A B^{2}+2 A C^{2}-B C^{2}=288+242-169=361=19^{2}
$$

Therefore, $A M=\frac{1}{2} A K=\frac{19}{2}$.

## Answer

$\frac{19}{2}$.