1. The problem states that triangle ABC is similar to triangle FGH, and we need to find the proportion to calculate the length of segment BC in centimeters. 2. Similar triangles have corresponding sides in proportion. This means the ratio of one side in triangle ABC to the corresponding side in triangle FGH is equal for all pairs of corresponding sides. 3. Given sides: - Triangle ABC: AB = 4.5 cm, AC = 6 cm, BC = unknown hypotenuse. - Triangle FGH: FG = 6.75 cm, FH = 9 cm, GH = 11.25 cm. 4. The proportion given is: $$\frac{9}{\text{blue box}} = \frac{\text{blue box}}{BC}$$ 5. We identify the corresponding sides: - AC (6 cm) corresponds to FH (9 cm). - AB (4.5 cm) corresponds to FG (6.75 cm). - BC corresponds to GH (11.25 cm). 6. Since 9 is FH, the first blue box must be AC = 6. 7. The second blue box corresponds to GH = 11.25. 8. So the proportion is: $$\frac{9}{6} = \frac{11.25}{BC}$$ 9. To find BC, cross multiply: $$9 \times BC = 6 \times 11.25$$ 10. Simplify the right side: $$9 \times BC = 67.5$$ 11. Divide both sides by 9: $$\cancel{9} \times BC = \frac{67.5}{\cancel{9}}$$ 12. Calculate: $$BC = 7.5$$ 13. Therefore, the length of segment BC is 7.5 cm. Final proportion: $$\frac{9}{6} = \frac{11.25}{BC}$$