B1. Let point $C(5,-2)$ be the midpoint of segment $A B$. The endpoints of the segment are points $A(2 a-b, a+b+1)$ and $B(3 a+2 b, 3 a+2 b-2)$, where $a$ and $b$ are real numbers. Calculate the length of segment $A B$. Express the result in simplified radical form.
B1. The point $S\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\right)$ is the midpoint of the segment between $A\left(x_{1}, y_{1}\right)$ and $B\left(x_{2}, y_{2}\right)$. This gives us the equations $\frac{5 a+b}{2}=5$ and $\frac{4 a+3 b-1}{2}=-2$. The solution to the system is $a=3$ and $b=-5$. For point $A$, we get the coordinates $A(11,-1)$ and for $B$, we get $B(-1,-3)$. Considering $d(A, B)=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}$, we get $d(A, B)=\sqrt{148}=2 \sqrt{37}$.

Considering or writing the coordinates of the midpoint of the segment $S\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\right) \ldots \ldots \ldots \ldots \ldots \ldots \ldots . \ldots . \ldots$ point

![](https://cdn.mathpix.com/cropped/2024_06_07_5bf9d67aee417dc7ba6ag-18.jpg?height=57&width=1727&top_left_y=1419&top_left_x=176)

![](https://cdn.mathpix.com/cropped/2024_06_07_5bf9d67aee417dc7ba6ag-18.jpg?height=60&width=1727&top_left_y=1466&top_left_x=176)

Solving the system .......................................................................................................................................

![](https://cdn.mathpix.com/cropped/2024_06_07_5bf9d67aee417dc7ba6ag-18.jpg?height=48&width=1727&top_left_y=1575&top_left_x=176)

![](https://cdn.mathpix.com/cropped/2024_06_07_5bf9d67aee417dc7ba6ag-18.jpg?height=49&width=1727&top_left_y=1620&top_left_x=176)

Considering or writing the formula for the length of the segment $d(A, B)=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}} \ldots 1$ point Calculating the length and partial simplification $d(A, B)=\sqrt{148}=2 \sqrt{37} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . \ldots$ point