7. The distance from the point of intersection of the diameter of a circle with a chord of length 18 cm to the center of the circle is $7 \mathrm{~cm}$. This point divides the chord in the ratio $2: 1$. Find the radius.

$$
A B=18, E O=7, A E=2 B E, R=?
$$
Solution: $2 B E \cdot B E=(R-7)(7+R)$

$$
A E \cdot B E=D E \cdot E C, \quad A E+B E=18, \quad B E=6
$$

$$
2 \cdot 6 \cdot 6=\left(R^{2}-7^{2}\right), R^{2}=72+49=121=11^{2}
$$

![](https://cdn.mathpix.com/cropped/2024_05_06_8eeeb8d6ace591217257g-17.jpg?height=322&width=380&top_left_y=313&top_left_x=1523)

Answer: $R=11$.