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The problem you're working on uses the **Pythagorean Theorem**, which states that for a right triangle, the relationship between the sides is:

\[
a^2 + b^2 = c^2
\]

Where:
- \(a\) and \(b\) are the legs of the triangle.
- \(c\) is the hypotenuse (the side opposite the right angle).

In your case:
- \(a = 5\)
- \(b = 8\)

Let's apply the Pythagorean theorem to find \(c\).

\[
c^2 = a^2 + b^2 = 5^2 + 8^2 = 25 + 64 = 89
\]

Now, take the square root of both sides to solve for \(c\):

\[
c = \sqrt{89} \approx 9.43
\]

So, the value of \(c\) is approximately **9.43**.

Would you like more details or have any questions?

Here are 5 related questions to expand on this:

1. How would the value of \(c\) change if \(b = 10\) but \(a = 5\) remains the same?
2. Can you explain why the Pythagorean theorem only applies to right triangles?
3. How do you calculate the length of the other side if you know the hypotenuse and one leg?
4. What is the significance of the square root function in the Pythagorean theorem?
5. How could you use the Pythagorean theorem to solve real-world problems involving right angles?

**Tip**: Memorizing common right triangles (like 3-4-5) can help you quickly identify side lengths in simple cases.

Learn how to solve for the hypotenuse (c) in a right triangle using the Pythagorean Theorem, given that a = 5 and b = 8. This problem is suitable for students in Grades 6-8 and covers essential geometry concepts.

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