1. **Problem statement:** We need to find the length of segment $AB$ in a triangular prism where $FE=8$ cm, $EC=14$ cm, the angle at vertex $F$ between edges $FA$ and $FB$ is $23^\circ$, and there is a right angle at vertex $A$ between edges $AF$ and $AB$. 2. **Understanding the geometry:** Since $AB$ is perpendicular to $AF$ at $A$, triangle $AFB$ is a right triangle with right angle at $A$. 3. **Using the angle at $F$:** The angle between $FA$ and $FB$ is $23^\circ$. Since $AB$ is perpendicular to $AF$, $AB$ is opposite the $23^\circ$ angle in triangle $AFB$. 4. **Applying trigonometry:** In right triangle $AFB$, $AB$ is opposite the $23^\circ$ angle, and $FB$ is the hypotenuse. We use the sine function: $$\sin(23^\circ) = \frac{AB}{FB}$$ 5. **Finding $FB$:** The length $FB$ corresponds to the edge $FE$ which is $8$ cm (assuming $FB=FE$ as per prism edges). 6. **Calculate $AB$:** $$AB = FB \times \sin(23^\circ) = 8 \times \sin(23^\circ)$$ 7. **Evaluate sine:** $$\sin(23^\circ) \approx 0.3907$$ 8. **Final calculation:** $$AB = 8 \times 0.3907 = 3.1256$$ 9. **Rounding to 2 decimal places:** $$AB \approx 3.13$$ **Answer:** The length of $AB$ is approximately $3.13$ cm.