1. **Problem Statement:** Given points on a number line and segments labeled with lengths and angles, we want to analyze the geometric relationships and calculate distances involving $\sqrt{56}$, $3+\sqrt{22}$, $3+\sqrt{13}$, and $\sqrt{57}$. 2. **Understanding the Setup:** The number line extends from 0 to 8 with points $O$ at 3, $B$ at 7, and $A$ at 8. Point $D$ is above $O$ connected by segment $\gamma$, and $C$ lies vertically below $D$ connected to $O$ also by $\gamma$. From $C$, a perpendicular segment $r'$ goes down to $B$. The segment $BA$ is horizontal on the number line. 3. **Key Formulas and Rules:** - Distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ - Pythagorean theorem for right triangles: $a^2 + b^2 = c^2$ - The length $r'$ is perpendicular from $C$ to $B$, so it forms a right angle. 4. **Calculations:** - Calculate $\sqrt{56} = \sqrt{7 \times 8} \approx 7.483$ (approximate length related to segment involving $D$ or $C$). - Calculate $3 + \sqrt{22} \approx 3 + 4.690 = 7.690$ (sum of 3 and the root, possibly a coordinate or length). - Calculate $3 + \sqrt{13} \approx 3 + 3.606 = 6.606$. - Calculate $\sqrt{57} \approx 7.550$. 5. **Interpretation:** These values likely represent distances or coordinates of points $D$, $C$, $B$, and $A$ relative to $O$ on the number line and vertical segments. The perpendicular $r'$ and segments $\gamma$ help form right triangles to apply the Pythagorean theorem. 6. **Summary:** Using the Pythagorean theorem and distance formula, the given expressions correspond to lengths of segments in the geometric figure, confirming the relationships between points $O$, $D$, $C$, $B$, and $A$ on the number line and vertical lines. **Final answer:** The distances are approximately $\sqrt{56} \approx 7.483$, $3+\sqrt{22} \approx 7.690$, $3+\sqrt{13} \approx 6.606$, and $\sqrt{57} \approx 7.550$.