Find the length of the third side. If necessary, write in simplest radical form.
5 and 5√2

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.

Find the length of the third side. If necessary, write in simplest radical form. 5 and 5√2

Learn how to find the hypotenuse of a right triangle with sides 5 and 5√2 using the Pythagorean Theorem. This step-by-step solution covers simplifying radicals and applying the theorem. Suitable for students in Grades 8-10.

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