B2 The point $S$ lies on the chord $A B$ of a circle such that $S A=3$ and $S B=5$. The radius of the circle from the center $M$ through $S$ intersects the circle at $C$. Given $C S=1$.

Calculate the length of the radius of the circle.

![](https://cdn.mathpix.com/cropped/2024_04_17_3de5ef26943dd9bef1e9g-2.jpg?height=411&width=640&top_left_y=1279&top_left_x=1116)
B2. Draw the perpendicular from $\mathrm{M}$ to the chord $\mathrm{AB}$.

Let the foot of the perpendicular be $T$.

$T$ is the midpoint of $AB$.

Let the length of the radius be $r$.

$A T^{2}+T M^{2}=r^{2}$, so $16+T M^{2}=r^{2}$

$S T^{2}+T M^{2}=S M^{2}$, so $1+T M^{2}=(r-1)^{2}$

(1) - (2) gives $15=r^{2}-(r-1)^{2}=2 r-1$, so $r=8$.

![](https://cdn.mathpix.com/cropped/2024_04_17_4ac1047f88758396ef6dg-1.jpg?height=329&width=466&top_left_y=2194&top_left_x=1229)