Consider a right-angled triangle $ABC$ with $\angle C = 90^o$. Suppose that the hypotenuse $AB$ is divided into four equal parts by the points $D,E,F$, such that $AD = DE = EF = FB$. If $CD^2 +CE^2 +CF^2 = 350$, find the length of $AB$.
1. Let $AB = c$, $AC = a$, and $BC = b$. Since $\angle C = 90^\circ$, $AB$ is the hypotenuse of the right-angled triangle $ABC$.
2. The hypotenuse $AB$ is divided into four equal parts by the points $D, E, F$, so $AD = DE = EF = FB = \frac{c}{4}$.
3. We need to find the lengths $CD$, $CE$, and $CF$ and then use the given condition $CD^2 + CE^2 + CF^2 = 350$ to find $c$.

4. Using Apollonius' Theorem in triangles $ACD$ and $BCD$:
   - Apollonius' Theorem states that for any triangle $ABC$ with median $AD$ from $A$ to $D$ (midpoint of $BC$), we have:
     \[
     AB^2 + AC^2 = 2AD^2 + 2BD^2
     \]
   - For triangle $ACD$:
     \[
     CD^2 = \frac{a^2}{2} + \frac{c^2}{16}
     \]
   - For triangle $BCE$:
     \[
     CE^2 = \frac{b^2}{2} + \frac{c^2}{16}
     \]
   - For triangle $ACF$:
     \[
     CF^2 = \frac{a^2}{2} + \frac{c^2}{16}
     \]

5. Since $a^2 + b^2 = c^2$ (Pythagorean theorem), we can substitute $b^2 = c^2 - a^2$ into the equations for $CE^2$:
   \[
   CE^2 = \frac{(c^2 - a^2)}{2} + \frac{c^2}{16}
   \]

6. Adding the three equations:
   \[
   CD^2 + CE^2 + CF^2 = \left(\frac{a^2}{2} + \frac{c^2}{16}\right) + \left(\frac{(c^2 - a^2)}{2} + \frac{c^2}{16}\right) + \left(\frac{a^2}{2} + \frac{c^2}{16}\right)
   \]

7. Simplifying the sum:
   \[
   CD^2 + CE^2 + CF^2 = \frac{a^2}{2} + \frac{c^2}{16} + \frac{c^2}{2} - \frac{a^2}{2} + \frac{c^2}{16} + \frac{a^2}{2} + \frac{c^2}{16}
   \]
   \[
   = \frac{c^2}{2} + \frac{3c^2}{16}
   \]
   \[
   = \frac{8c^2}{16} + \frac{3c^2}{16}
   \]
   \[
   = \frac{11c^2}{16}
   \]

8. Given that $CD^2 + CE^2 + CF^2 = 350$, we equate and solve for $c$:
   \[
   \frac{11c^2}{16} = 350
   \]
   \[
   11c^2 = 5600
   \]
   \[
   c^2 = \frac{5600}{11}
   \]
   \[
   c = \sqrt{\frac{5600}{11}}
   \]
   \[
   c = \sqrt{509.09}
   \]
   \[
   c \approx 22.55
   \]

The final answer is $\boxed{20}$