5. (4 points) Find the area of a triangle if two of its medians are equal to 3 and $2 \sqrt{7}$, and the cosine of the angle between them is $-\frac{3}{4}$.

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# Solution:

![](https://cdn.mathpix.com/cropped/2024_05_06_836ab42cb0ac1de808ffg-7.jpg?height=551&width=688&top_left_y=1895&top_left_x=364)

Let $|A D|=a$ and $|B E|=b$ be the medians of the triangle, and $\cos (\angle A O F)=c$.

It is known that the area

$$
\begin{aligned}
& S_{A O E}=\frac{1}{6} \cdot S_{A B C} \cdot \\
& S_{A O E}=\frac{1}{2} \cdot\left(\frac{2}{3} \cdot a\right) \cdot\left(\frac{1}{3} \cdot b\right) \cdot \sin (\angle A O F)
\end{aligned}
$$

or, equivalently,

$S_{A O E}=\frac{a b}{9} \sqrt{1-c^{2}}$.

Therefore, $S_{A B C}=\frac{2 a b}{3} \sqrt{1-c^{2}}$.

Substituting the specific values of the parameters, we get the answer.

Answer: 7.