10.017. Given a point $P$, which is 7 cm away from the center of a circle with a radius of $11 \mathrm{~cm}$. A chord of length 18 cm is drawn through this point. What are the lengths of the segments into which the chord is divided by point $P$?
## Solution.

Draw the diameter $C D$ through point $P$ (Fig. 10.17), which will divide it into segments $P D$ and $C P$ of lengths $11-7=4$ and $11+7=18$ (cm). Let $A P=x$; then $P B=18-x$. Since $A P \cdot P B=C P \cdot P D=4 \cdot 18$, we have $x(18-x)=72$ or $x^{2}-18 x+72=0$, from which $x_{1}=12, x_{2}=6$, i.e., the chord $A B$ is divided by point $P$ into segments of lengths 12 and 6 cm.

Answer: 12 and 6 cm.