\left.\begin{array}{l}{[\text { Product of the lengths of chord segments and secant segments ] }} \\ {[\quad \text { Diameter, main properties }}\end{array}\right]

Point $P$ is 7 units away from the center of a circle with a radius of 11. A chord passing through point $P$ is 18 units long. Find the segments into which the chord is divided by point $P$?
Drop a perpendicular from the center of the circle to the given chord.

## Solution

## First method.

Let $O$ be the center of the given circle, $AB$ be the given chord, $AB=18, OP=7$. Let $M$ be the foot of the perpendicular dropped from the center $O$ to the chord $AB$. Then $M$ is the midpoint of $AB$ and

$$
OM=\sqrt{OA^2-AM^2}=\sqrt{40}
$$

If point $P$ is between $M$ and $B$, then

$$
PM=\sqrt{OP^2-OM^2}=3
$$

Second method.

Let $O$ be the center of the circle, $AB$ be the given chord. Draw the diameter $CD$ passing through point $P$ ($P$ is between $O$ and $D$). Let $PB=x$. Then

$$
\begin{gathered}
AP=18-x, DP=OD-OP=11-7=4 \\
PC=OP+OC=7+11=18, AP \cdot PB=PD \cdot PC
\end{gathered}
$$

or

$$
(18-x) x=4 \cdot 18
$$

From this equation, we find that $x=12$ or $x=6$.

![](https://cdn.mathpix.com/cropped/2024_05_06_36c4956010bd91c4b4eeg-05.jpg?height=497&width=489&top_left_y=532&top_left_x=790)

Answer

12 and 6.