2. The area of triangle $ABC$ with vertices $A(-1,-6), B(1,0), C(3,-3)$ is 9. Calculate the length of the median to side $b$ and the distance between the foot of the altitude on side $b$ and the midpoint of side $b$.
2. We calculate the coordinates of the midpoint of side $AC$, which is $R\left(\frac{-1+3}{2}, \frac{-3-6}{2}\right)$. We calculate $t_{b}=$ $d(B, R)=\frac{9}{2}$. We calculate the length of side $b=d(A, C)=5$. We use the formula $S=\frac{b \cdot v_{b}}{2}$ and get $v_{b}=\frac{18}{5}$. We consider the relationship $t_{b}^{2}=v_{b}^{2}+x^{2}$, where $x$ is the sought distance. This measures $x=\frac{27}{10}$.

Calculated coordinates of the midpoint $R\left(1,-\frac{9}{2}\right) \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots$

![](https://cdn.mathpix.com/cropped/2024_06_07_2b0bf3255e176946d93cg-08.jpg?height=60&width=1639&top_left_y=507&top_left_x=206)

![](https://cdn.mathpix.com/cropped/2024_06_07_2b0bf3255e176946d93cg-08.jpg?height=57&width=1639&top_left_y=565&top_left_x=206)

![](https://cdn.mathpix.com/cropped/2024_06_07_2b0bf3255e176946d93cg-08.jpg?height=51&width=1642&top_left_y=614&top_left_x=207)

![](https://cdn.mathpix.com/cropped/2024_06_07_2b0bf3255e176946d93cg-08.jpg?height=60&width=1642&top_left_y=661&top_left_x=207)

![](https://cdn.mathpix.com/cropped/2024_06_07_2b0bf3255e176946d93cg-08.jpg?height=65&width=1639&top_left_y=710&top_left_x=206)