1. **State the problem:** We have two right triangles DBA and EBD sharing vertex D, with right angles at D. Given lengths are $EA=\sqrt{13}$ cm, $DE=2.5$ cm, and $BA=3$ cm. We need to find the length $DB$. 2. **Understand the setup:** Since $D$ is the right angle in both triangles, and $EBD$ is inside $DBA$, the triangles are similar by AA similarity (both have right angles and share angle at B). 3. **Use similarity ratios:** For similar triangles $DBA$ and $EBD$, corresponding sides are proportional: $$\frac{DB}{EB} = \frac{BA}{BD} = \frac{DA}{ED}$$ 4. **Express known lengths:** We know $BA=3$ cm, $EA=\sqrt{13}$ cm, and $DE=2.5$ cm. Note that $EA = ED + DA$, so $$DA = EA - ED = \sqrt{13} - 2.5$$ 5. **Set variables:** Let $DB = x$ (the length to find), and $EB = y$. 6. **From similarity:** $$\frac{x}{y} = \frac{3}{x} \implies x^2 = 3y$$ 7. **Use Pythagoras in triangle EBD:** $$EB^2 = ED^2 + DB^2 \implies y^2 = 2.5^2 + x^2 = 6.25 + x^2$$ 8. **Substitute $y$ from step 6:** $$y = \frac{x^2}{3}$$ 9. **Plug into Pythagoras:** $$\left(\frac{x^2}{3}\right)^2 = 6.25 + x^2$$ $$\frac{x^4}{9} = 6.25 + x^2$$ 10. **Multiply both sides by 9:** $$x^4 = 56.25 + 9x^2$$ 11. **Rearrange:** $$x^4 - 9x^2 - 56.25 = 0$$ 12. **Let $z = x^2$:** $$z^2 - 9z - 56.25 = 0$$ 13. **Solve quadratic for $z$:** $$z = \frac{9 \pm \sqrt{81 + 225}}{2} = \frac{9 \pm \sqrt{306}}{2}$$ 14. **Calculate discriminant:** $$\sqrt{306} \approx 17.4929$$ 15. **Find roots:** $$z_1 = \frac{9 + 17.4929}{2} = 13.2465, \quad z_2 = \frac{9 - 17.4929}{2} = -4.2465$$ 16. **Discard negative root since $z = x^2 \geq 0$:** $$x^2 = 13.2465 \implies x = \sqrt{13.2465} \approx 3.64$$ **Final answer:** $$\boxed{DB \approx 3.64 \text{ cm}}$$