1. **State the problem:** We have a right triangle ABC with the right angle at A. Point D lies on AC such that $AD=3$ cm and $DC=5$ cm. Point E lies on CB such that $CE=EB$, and segment DE is perpendicular to CB. We need to find the length $EC$. 2. **Analyze the triangle:** Since $AD=3$ and $DC=5$, the total length of AC is $3+5=8$ cm. 3. **Right triangle properties:** Triangle ABC is right-angled at A, so by the Pythagorean theorem: $$AB^2 + AC^2 = BC^2$$ We know $AC=8$, but $AB$ and $BC$ are unknown. 4. **Position of E on CB:** Since $CE=EB$, E is the midpoint of CB. So, $$E = \text{midpoint of } CB$$ 5. **Perpendicular from D to CB:** DE is perpendicular to CB. 6. **Coordinate geometry approach:** Let's place point A at the origin $(0,0)$. Since AC lies along the x-axis, let: $$A = (0,0), C = (8,0)$$ Point D divides AC with $AD=3$, so: $$D = (3,0)$$ 7. **Coordinates of point B:** Since triangle ABC is right-angled at A, AB is vertical to AC (x-axis), so B lies on the y-axis: $$B = (0,h)$$ for some $h$. 8. **Coordinates of point E (midpoint of CB):** $$C = (8,0), B = (0,h)$$ $$E = \left( \frac{8+0}{2}, \frac{0+h}{2} \right) = \left(4, \frac{h}{2}\right)$$ 9. **Line CB:** Slope of CB is: $$m_{CB} = \frac{h-0}{0-8} = \frac{h}{-8} = -\frac{h}{8}$$ 10. **Equation of CB:** Using point C: $$y - 0 = m_{CB}(x - 8)$$ $$y = -\frac{h}{8}(x - 8)$$ 11. **Equation of DE:** DE is perpendicular to CB, so slope of DE is: $$m_{DE} = -\frac{1}{m_{CB}} = -\frac{1}{-h/8} = \frac{8}{h}$$ 12. **Line DE passes through D (3,0):** $$y - 0 = m_{DE}(x - 3)$$ $$y = \frac{8}{h}(x - 3)$$ 13. **Point E is on both CB and DE:** At $x=4$ for E, so from DE equation: $$y = \frac{8}{h}(4 - 3) = \frac{8}{h}$$ And from E coordinates: $$y = \frac{h}{2}$$ Set equal: $$\frac{8}{h} = \frac{h}{2}$$ Multiply both sides by $2h$: $$16 = h^2$$ $$h = 4 \quad \text{(taking positive length)}$$ 14. **Find length EC:** Since $E = (4, h/2) = (4, 2)$ and $C=(8,0)$, $$EC = \sqrt{(8-4)^2 + (0-2)^2} = \sqrt{4^2 + (-2)^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5}$$ cm. **Final answer:** $EC = 2\sqrt{5}$ cm. Therefore, the correct choice is (c) 2√5.