Problem 9.5. Point $M$ is the midpoint of side $B C$ of triangle $A B C$, where $A B=17$, $A C=30, B C=19$. A circle is constructed with side $A B$ as its diameter. An arbitrary point $X$ is chosen on this circle. What is the minimum value that the length of segment $M X$ can take?

![](https://cdn.mathpix.com/cropped/2024_05_06_790dd471148872cd0846g-03.jpg?height=414&width=553&top_left_y=1517&top_left_x=448)
Answer: 6.5.

Solution. For the segment $M X$ to be minimal, we need the point $X$ to lie on the segment $M O$, where $O$ is the center of the circle (and also the midpoint of side $A B$), as shown in Fig. 2.

![](https://cdn.mathpix.com/cropped/2024_05_06_790dd471148872cd0846g-04.jpg?height=430&width=555&top_left_y=310&top_left_x=449)

Fig. 2: to the solution of problem 9.5

It remains to note that $O M=\frac{1}{2} A C$ (this is the midline), and $O X=\frac{1}{2} A B$ (this is the radius of the circle constructed on $A B$ as the diameter). We get

$$
M X=O M-O X=\frac{1}{2}(A C-A B)=6.5
$$