1. **Problem statement:** Find the angle $\angle RPQ$ and the perpendicular distance from point R to line PQ in the triangle formed by the Leaning Tower of Pisa. 2. **Given data:** - Height of tower (RP) = 184.5 ft - Distance from base (PQ) = 123 ft - Angle of elevation at Q = 60° 3. **Find:** - $\angle RPQ$ - Perpendicular distance from R to PQ 4. **Step 1: Understand the triangle** - Triangle RPQ is right-angled at P (since RP is vertical height and PQ is horizontal base). - $\angle RQP = 60^\circ$ is given. 5. **Step 2: Use trigonometric ratios** - We know $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$. - Here, $\theta = 60^\circ$, opposite side = RP = 184.5 ft, adjacent side = PQ = 123 ft. 6. **Step 3: Calculate $\tan(60^\circ)$** $$\tan(60^\circ) = \sqrt{3} \approx 1.732$$ 7. **Step 4: Check if given sides satisfy the tangent ratio** $$\tan(60^\circ) = \frac{RP}{PQ} = \frac{184.5}{123} \approx 1.5$$ - Since 1.5 \neq 1.732, the triangle is not right angled at P, so $\angle RPQ$ is not 90°. 8. **Step 5: Use Law of Cosines or Law of Sines to find $\angle RPQ$** - Let $\angle RPQ = x$. - Using Law of Cosines in triangle RPQ: $$RQ^2 = RP^2 + PQ^2 - 2 \times RP \times PQ \times \cos(x)$$ 9. **Step 6: Find length RQ using angle at Q** - Using Law of Sines: $$\frac{RP}{\sin(\angle PQ R)} = \frac{RQ}{\sin(60^\circ)}$$ - But $\angle PQ R = 90^\circ - x$ because angles in triangle sum to 180° and $\angle P = 90^\circ$. 10. **Step 7: Calculate RQ using Pythagoras** $$RQ = \sqrt{RP^2 + PQ^2} = \sqrt{184.5^2 + 123^2} = \sqrt{34052.25 + 15129} = \sqrt{49181.25} \approx 221.8 \text{ ft}$$ 11. **Step 8: Use Law of Sines to find $x$** $$\frac{RP}{\sin(90^\circ - x)} = \frac{RQ}{\sin(60^\circ)}$$ $$\Rightarrow \frac{184.5}{\cos(x)} = \frac{221.8}{0.866}$$ $$\Rightarrow 184.5 \times 0.866 = 221.8 \times \cos(x)$$ $$\Rightarrow 159.7 = 221.8 \times \cos(x)$$ $$\Rightarrow \cos(x) = \frac{159.7}{221.8} \approx 0.72$$ 12. **Step 9: Calculate $x$** $$x = \cos^{-1}(0.72) \approx 43.96^\circ$$ 13. **Step 10: Find perpendicular distance from R to PQ** - The perpendicular distance is the height RP = 184.5 ft because RP is vertical height from R to PQ. **Final answers:** - $\angle RPQ \approx 43.96^\circ$ - Perpendicular distance from R to PQ = 184.5 ft