8,9 In a circle of radius $R$, a chord equal to $R / 2$ is drawn. Through one end of the chord, a tangent to the circle is drawn, and through the other end - a secant parallel to the tangent. Find the distance between the tangent and the secant.
Draw a diameter through the point of tangency.

## Solution

Let $AB$ be the given chord, $BC$ be the segment of the given secant inside the circle, and $M$ be the point of intersection of $AD$ and $BC$. Draw the diameter $AD$ through the point of tangency $A$. Then $BC \perp AD$ and $CM = MB$. From the right triangle $ABD$, we find that $AM \cdot AD = AB^2$, $AM = \frac{AB^2}{AD} = \frac{R}{8}$.

![](https://cdn.mathpix.com/cropped/2024_05_06_0ef0b80dacb6d98eff5ag-11.jpg?height=574&width=644&top_left_y=-1&top_left_x=707)

## Answer