2. Let $D$ be the midpoint of side $A B$, and $T$ the centroid of triangle $A B C$. Calculate the lengths of its sides if $|A D|=3,|D T|=5$ and $|T A|=4$.


IV/2. Point $D$ is the midpoint of side $AB$ and since $|AD|=3$, it follows that $|AB|=6$. Let $E$ be the midpoint of side $BC$ and $F$ the midpoint of $AC$. In triangle $ADT$, the Pythagorean theorem holds for the side lengths, so this triangle is a right triangle. Since the centroid divides the median in the ratio 2 to 1 and $|AT|=4$, it follows that $|AE|=6$. Therefore, in the right triangle $ABE$, we know the lengths of the legs, so we can calculate $|BE|=\sqrt{|AB|^2 + |AE|^2} = \sqrt{36 + 36} = 6\sqrt{2}$. From this, it follows that $|BC|=2|BE|=12\sqrt{2}$.

Since $E$ and $F$ are the midpoints of sides $BC$ and $AC$, $EF$ is parallel to $AB$. Since $EA$ is perpendicular to $AB$, $EA$ is also perpendicular to $EF$. Therefore, triangle $AEF$ is a right triangle.

![](https://cdn.mathpix.com/cropped/2024_06_07_24964e3bc413fb5af60fg-16.jpg?height=602&width=536&top_left_y=1635&top_left_x=1314)
We already know that $|AE|=6$, and it also holds that $|EF|=\frac{|AB|}{2}=3$. Therefore, we can calculate the length of $FA$ using the Pythagorean theorem as follows:

$$
|FA|=\sqrt{|EF|^2 + |AE|^2} = \sqrt{9 + 36} = 3\sqrt{5}
$$

From this, it follows that the length of side $AC$ is $|AC|=2|AF|=6\sqrt{5}$.

Conclusion $|AB|=6$

Observation that triangle $ADT$ is a right triangle
Calculation of the length $|BC|=12\sqrt{2}$ ..... 2 points
Calculation of the length $|AC|=6\sqrt{5}$ ..... 2 points