[

doubling the median ] [Theorem of the sum of the squares of the diagonals ]

In a triangle, two sides are equal to 11 and 23, and the median drawn to the third side is 10. Find the third 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

First method.

Let $C M$ be the median of triangle $A B C$, where $A C=11, B C=23, C M=10$. Then

$$
C M^{2}=\frac{1}{4}\left(2 A C^{2}+2 B C^{2}-A B^{2}\right) \text {, or } 100=\frac{1}{4}\left(2 \cdot 121+2 \cdot 529-A B^{2}\right) .
$$

From this, we find that $A B^{2}=900$.

Let $C M$ be the median of triangle $A B C$, where $A C=11, B C=23, C M=10$. On the extension of the median $C M$ beyond point $M$, lay off segment $M D$ equal to $C M$. Then $A C B D$ is a parallelogram, $C D=20, D B=11$. Find $\cos \angle C D B$ from triangle $C D B$, and then - segment $A M$ from triangle $A C M$ ( $\angle A C M=\angle C D B$ ).

## Answer

30.