[ Properties and characteristics of tangents ] [ Pythagorean Theorem (direct and inverse) ]

Given two parallel lines at a distance of 15 from each other; between them is a point $M$ at a distance of 3 from one of them. A circle is drawn through point $M$, touching both lines. Find the distance between the projections of the center and point $M$ on one of the given lines.
Drop a perpendicular from point $M$ to the radius of the circle, drawn to one of the points of tangency.

## Solution

Let $A$ and $B$ be the projections of point $M$ and the center $O$ of the circle onto one of the lines, with $M A=3$. Then $O B$ is the radius of the circle and $O B=\frac{15}{2}$.

Let $P$ be the projection of point $M$ onto $O B$. In the right triangle $M P O$, it is known that

$$
M O=\frac{15}{2}, O P=O B-P B=O B-M A=\frac{15}{2}-3=\frac{9}{2}
$$

Therefore,

$$
A B^{2}=M P^{2}=M O^{2}-O P^{2}=\left(\frac{15}{2}\right)^{2}-\left(\frac{9}{2}\right)^{2}=36
$$

![](https://cdn.mathpix.com/cropped/2024_05_06_02603d12f3437ea52c37g-26.jpg?height=372&width=669&top_left_y=363&top_left_x=703)

## Answer

6.