$[$ Theorem on the lengths of a tangent and a secant; the product of the entire secant and its external part

From a point $M$, located outside the circle at a distance of $\sqrt{7}$ from the center, a secant is drawn, the internal part of which is half the external part and equal to the radius of the circle.

Find the radius of the circle.
Apply the tangent-secant theorem.

## Solution

Let $r$ be the radius of the circle. According to the tangent-secant theorem, $2 r \cdot 3 r = (\sqrt{7} - r)(\sqrt{7} + r) = 7 - r^2$, from which we get $r = 1$.

## Answer

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Point $B$ lies on segment $AC$, which is equal to 5. Find the distance between the midpoints of segments $AB$ and $BC$.

## Hint

The distance between the midpoints of the given segments is half the length of segment $AC$.

## Solution

Let $M$ and $N$ be the midpoints of segments $AB$ and $BC$, respectively. Then segment $MN$ is composed of half of segments $AB$ and $BC$, and since $AB + BC = 5$, we have

$MN = \frac{1}{2} AB + \frac{1}{2} BC = \frac{1}{2} AC = \frac{5}{2}$.

## Answer

2.5.