1. On a plane, concentric circles with radii of $7 \mathrm{~cm}$ and $11 \mathrm{~cm}$ are drawn. The smaller circle divides a chord of the larger circle into 3 equal parts. How long is this chord?
$\mathbf{I} / 1$. Let the endpoints of the chord be $A$ and $B$, and the intersection points of the chord with the smaller circle be $C$ and $D$. Connect points $A$ and $C$ to the center $O$ of the circles. Draw a perpendicular from the center $O$ to the chord and denote its foot by $N$. We know that $|A C|=|C D|=|D B|$ and that point $N$ bisects segment $C D$. Let $|C N|=x$. Then $|A C|=2 x$. By the Pythagorean theorem, $11^{2}-(3 x)^{2}=7^{2}-x^{2}$, from which we get $8 x^{2}=72$. Since we are interested in the positive solution, we choose $x=3$. The chord $A B$ is $18 \mathrm{~cm}$ long.

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Introduction of 2 right triangles with a common leg (e.g., $A O N$ and $C O N$) ... 1 point The condition $|A C|=|C D|=|D B|$ (or equivalent) ..................... 1 point Expressing the length $|O N|$ using the Pythagorean theorem in $A O N$ or $C O N \ldots \ldots .1$ point

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