1. **Problem Statement:** We have a circle with points L and G on the circumference. A chord FG passes through the circle and is intersected by line LS at point S. The chord FG is divided into segments FS = 6 and SA = 4. The segment LS = 3 and SG = x. We need to find the value of $x$. 2. **Relevant Theorem:** When two chords intersect inside a circle, the products of the lengths of the segments of each chord are equal. This is called the Intersecting Chords Theorem. 3. **Formula:** If two chords intersect at point S, then: $$LS \times SG = FS \times SA$$ 4. **Substitute known values:** $$3 \times x = 6 \times 4$$ 5. **Calculate:** $$3x = 24$$ 6. **Solve for $x$:** $$x = \frac{24}{3} = 8$$ 7. **Answer:** The length of segment $SG$ is $8$. This means the segment from S to G on the chord FG is 8 units long.