1. **State the problem:** We need to find the length of segment $AB$ in the given geometric figure. 2. **Analyze the figure:** The figure has two right angles at the bottom corners, vertical sides of lengths 8.7 and 11.5, and a top horizontal segment of length 10.4. There is an angle of $31.4^\circ$ between lines connecting points $A$ and $B$. 3. **Identify the triangle and apply the Law of Cosines:** To find $AB$, we consider the triangle formed by points $A$, $B$, and the bottom left corner. The sides adjacent to the angle $31.4^\circ$ are the vertical difference $|11.5 - 8.7| = 2.8$ and the horizontal segment $10.4$. 4. **Law of Cosines formula:** $$AB^2 = 10.4^2 + 2.8^2 - 2 \times 10.4 \times 2.8 \times \cos(31.4^\circ)$$ 5. **Calculate each term:** $$10.4^2 = 108.16$$ $$2.8^2 = 7.84$$ $$2 \times 10.4 \times 2.8 = 58.24$$ 6. **Calculate cosine:** $$\cos(31.4^\circ) \approx 0.8526$$ 7. **Substitute values:** $$AB^2 = 108.16 + 7.84 - 58.24 \times 0.8526 = 116 - 49.63 = 66.37$$ 8. **Find $AB$ by taking the square root:** $$AB = \sqrt{66.37} \approx 8.15$$ **Final answer:** $$\boxed{AB \approx 8.15}$$