In the right triangle ABC shown, E and D are the trisection points of the hypotenuse AB. If CD = 7 and CE = 6, what is the length of hypotenuse AB? 

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.

In the right triangle ABC shown, E and D are the trisection points of the hypotenuse AB. If CD = 7 and CE = 6, what is the length of hypotenuse AB?

Explore how to determine the length of hypotenuse AB in a right triangle ABC, where points E and D trisect the hypotenuse. Given CD = 7 and CE = 6, this solution applies the distance formula, Pythagorean theorem, and systems of equations for a step-by-step approach. Ideal for students in Grades 10-12.

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