To solve this problem, we can use the information about points $D$ and $E$ being trisection points on the hypotenuse $AB$ and apply the Pythagorean theorem with a coordinate-based approach.

Let's proceed with the following steps:

1. Assign coordinates to points:

   • Let $A$ be at the origin $(0, 0)$.
   • Let $B$ be at $(x, y)$.
   • Since $D$ and $E$ are trisection points, we can place $D$ at $\left(\frac{x}{3}, \frac{y}{3}\right)$ and $E$ at $\left(\frac{2x}{3}, \frac{2y}{3}\right)$.

2. Calculate lengths $CD$ and $CE$:

   • We know $CD = 7$ and $CE = 6$.
   • Using the distance formula, we can set up equations for each segment.

   For $CD$: $CD = \sqrt{\left(\frac{x}{3} - 0\right)^2 + \left(\frac{y}{3} - y\right)^2} = 7$ Simplifying, we get: $\sqrt{\left(\frac{x}{3}\right)^2 + \left(\frac{-2y}{3}\right)^2} = 7$ $\sqrt{\frac{x^2}{9} + \frac{4y^2}{9}} = 7$ Squaring both sides: $\frac{x^2 + 4y^2}{9} = 49$ $x^2 + 4y^2 = 441$

   For $CE$: $CE = \sqrt{\left(\frac{2x}{3} - 0\right)^2 + \left(\frac{2y}{3} - y\right)^2} = 6$ Simplifying, we get: $\sqrt{\left(\frac{2x}{3}\right)^2 + \left(\frac{-y}{3}\right)^2} = 6$ $\sqrt{\frac{4x^2}{9} + \frac{y^2}{9}} = 6$ Squaring both sides: $\frac{4x^2 + y^2}{9} = 36$ $4x^2 + y^2 = 324$

3. Solve the system of equations: We now have two equations: $x^2 + 4y^2 = 441$ $4x^2 + y^2 = 324$

   To eliminate one variable, we can multiply the first equation by 4: $4x^2 + 16y^2 = 1764$ Now, subtract the second equation: $(4x^2 + 16y^2) - (4x^2 + y^2) = 1764 - 324$ $15y^2 = 1440$ $y^2 = 96$ $y = \sqrt{96} = 4\sqrt{6}$

   Substitute $y = 4\sqrt{6}$ into the first equation: $x^2 + 4(96) = 441$ $x^2 + 384 = 441$ $x^2 = 57$ $x = \sqrt{57}$

4. Find the length of hypotenuse $AB$: The length of $AB$ is: $AB = \sqrt{x^2 + y^2} = \sqrt{57 + 96} = \sqrt{153} = 3\sqrt{17}$

Thus, the length of the hypotenuse $AB$ is $\boxed{3\sqrt{17}}$.

Would you like any further details or have questions on the solution steps?

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Here are 5 related questions to explore further:

1. How does the trisection of a segment affect the coordinate locations of points?
2. What other methods can solve a problem involving trisection points and distances?
3. How would this change if the triangle was not a right triangle?
4. Can this approach be applied if CD and CE were in different ratios?
5. How does the Pythagorean theorem apply in coordinate-based proofs?

Tip: Remember to break complex problems down by assigning coordinates to key points and setting up equations systematically.