10. On the hypotenuse $A B$ of the right triangle $A B C$, a square $A B D E$ is constructed in the half-plane that does not contain triangle $A B C$. Find the distance from the vertex $C$ of the right angle of the triangle to the center of the square, given that $B C=a$ and $A C=b$.
Solution 1. Introduce a coordinate system, the origin of which coincides with the vertex $C$ of the right angle of the triangle, and the ends of the hypotenuse lie on the coordinate axes at points $A(b ; 0)$ and $B(0 ; a)$. The point $E$ - one of the vertices of the square (left figure) - has coordinates $(a+b ; b)$.

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Therefore, the center of the square - point $M$ - has coordinates $\left(\frac{a+b}{2} ; \frac{a+b}{2}\right)$. Consequently, $C M=\frac{a+b}{\sqrt{2}}$.

Topic: "Coordinate Method"
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