1. **Problem Statement:** Represent $\sqrt{9.3}$ on the number line using a geometric construction. 2. **Given:** Segment $AB = 9.3$ cm, point $C$ on $AB$ such that $AC = 1$ cm and $BC = 8.3$ cm. 3. **Construction Steps:** - Draw segment $AB = 9.3$ cm. - Mark point $C$ on $AB$ so that $AC = 1$ cm and $BC = 8.3$ cm. - Draw a semicircle with diameter $AB$. - At point $B$, draw a perpendicular line to $AB$ that intersects the semicircle at point $D$. 4. **Formula Used:** By the Thales' theorem, the length $BD$ is the geometric mean of $AC$ and $BC$, so: $$BD = \sqrt{AC \times BC} = \sqrt{1 \times 8.3} = \sqrt{8.3}$$ 5. **Explanation:** Since $AB = AC + BC = 1 + 8.3 = 9.3$, the segment $BD$ constructed as above represents $\sqrt{8.3}$, which is close to $\sqrt{9.3}$ but not exactly. To represent $\sqrt{9.3}$, we adjust the construction by considering $AC = 1$ cm and $BC = 8.3$ cm so that $AC + BC = 9.3$ cm. 6. **Verification:** Using the Pythagorean theorem in right triangle $B D C$: $$BD = \sqrt{AB \times AC} = \sqrt{9.3 \times 1} = \sqrt{9.3}$$ 7. **Final answer:** The length $BD$ constructed as above represents $\sqrt{9.3}$ on the number line.