1. **Problem statement:** We have three lines AB, BC, and CD with given lengths and directions. A reversed curve connects these lines, and we need to find the length of the common radius of this reversed curve. 2. **Understanding the problem:** A reversed curve consists of two curves bending in opposite directions connected by a short tangent. The common radius is the radius of these curves. 3. **Given data:** - AB = 57.6 m due East - BC = 91.5 m at N 68° E - CD = 102.6 m at S 48° E 4. **Step 1: Calculate the bearings of BC and CD relative to East:** - BC bearing from East: 90° - 68° = 22° (since N 68° E means 68° from North towards East, so from East it is 90° - 68°) - CD bearing from East: 360° - 48° = 312° (since S 48° E means 48° from South towards East, so from East it is 360° - 48°) 5. **Step 2: Calculate the angle between BC and CD:** The angle between BC and CD is the difference between their bearings: $$\theta = 312^\circ - 22^\circ = 290^\circ$$ Since angles greater than 180° are reflex, the internal angle is: $$360^\circ - 290^\circ = 70^\circ$$ 6. **Step 3: Calculate the deflection angle between BC and CD:** The deflection angle $\Delta$ is the angle between the tangents of the reversed curve, which is $70^\circ$. 7. **Step 4: Use the formula for the length of the common radius $R$ of a reversed curve:** For a reversed curve with two equal radii and a short tangent $T$ between them, the length of the tangent $T$ is related to the radius $R$ and deflection angle $\Delta$ by: $$T = R \tan\left(\frac{\Delta}{2}\right)$$ 8. **Step 5: Calculate the length of the short tangent $T$:** The short tangent $T$ is the length of AB, which is 57.6 m. 9. **Step 6: Solve for $R$:** $$R = \frac{T}{\tan\left(\frac{\Delta}{2}\right)} = \frac{57.6}{\tan\left(\frac{70^\circ}{2}\right)} = \frac{57.6}{\tan(35^\circ)}$$ 10. **Step 7: Calculate $\tan(35^\circ)$:** $$\tan(35^\circ) \approx 0.7002$$ 11. **Step 8: Calculate $R$:** $$R = \frac{57.6}{0.7002} \approx 82.3 \text{ m}$$ **Final answer:** The length of the common radius of the reversed curve is approximately **82.3 m**.