1. **State the problem:** Pearl places a mirror 45 feet from the base of a building. She stands 12 feet from the mirror and can see the top of the building in the mirror. Pearl's height is 5 feet 9 inches. We need to find the height of the building. 2. **Convert Pearl's height to feet:** 5 feet 9 inches = 5 + \frac{9}{12} = 5.75 feet. 3. **Identify similar triangles:** The small triangle formed by Pearl and the mirror and the larger triangle formed by the building and the mirror are similar because the angles of incidence and reflection are equal. 4. **Set up the proportion:** Height of Pearl / Distance from Pearl to mirror = Height of building / Distance from building to mirror $$\frac{5.75}{12} = \frac{h}{45}$$ 5. **Solve for building height $h$:** Multiply both sides by 45: $$h = 45 \times \frac{5.75}{12}$$ 6. **Simplify the fraction:** $$h = 45 \times \frac{5.75}{\cancel{12}}$$ $$h = 45 \times \frac{5.75}{\cancel{12}}$$ 7. **Calculate:** $$h = 45 \times 0.479167 = 21.5625$$ 8. **Round to the nearest thousandth:** $$h \approx 21.563$$ feet. **Final answer:** The height of the building is approximately **21.563** feet.