To find the length of the third side of a triangle, you need to provide more information about the type of triangle (right triangle, isosceles, equilateral, etc.) or any other relevant details, like the angle between the sides or whether it's a right triangle.

For now, I'll assume this is a right triangle. If 5 and $5\sqrt{2}$ are the lengths of the legs of a right triangle, we can use the Pythagorean theorem to find the length of the hypotenuse:

$c = \sqrt{a^2 + b^2}$

where $a = 5$ and $b = 5\sqrt{2}$.

Substituting these values:

$c = \sqrt{5^2 + (5\sqrt{2})^2}$ $c = \sqrt{25 + 25 \times 2}$ $c = \sqrt{25 + 50}$ $c = \sqrt{75}$ $c = \sqrt{25 \times 3} = 5\sqrt{3}$

Thus, the length of the hypotenuse is $5\sqrt{3}$.

Would you like further details or have any questions?

Here are 5 questions for further exploration:

1. What is the Pythagorean theorem, and how is it used in right triangles?
2. Can you determine the length of a side if you only have two sides and an angle?
3. How would the solution change if the triangle were not a right triangle?
4. Can you simplify $\sqrt{75}$ further, and what are the steps involved?
5. How does the triangle's type (isosceles, equilateral) affect the calculation of the third side?

Tip: Always check if the sides correspond to the legs or the hypotenuse in right triangles before applying the Pythagorean theorem.