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The problem shows a right triangle with one leg of length \( 6 \), another leg of length \( 3\sqrt{13} \), and the goal is to find the length of the hypotenuse.

We can use the **Pythagorean Theorem** to solve this. The theorem states:

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

Where:
- \( c \) is the hypotenuse
- \( a \) and \( b \) are the two legs

Given:
- \( a = 6 \)
- \( b = 3\sqrt{13} \)

Let's calculate the hypotenuse \( c \):

\[
c^2 = 6^2 + (3\sqrt{13})^2
\]
\[
c^2 = 36 + 9 \times 13
\]
\[
c^2 = 36 + 117 = 153
\]
\[
c = \sqrt{153}
\]

Now, we simplify \( \sqrt{153} \). Notice that 153 can be factored as \( 153 = 9 \times 17 \), so:

\[
c = \sqrt{9 \times 17} = 3\sqrt{17}
\]

Thus, the length of the hypotenuse is \( 3\sqrt{17} \).

Would you like further explanation or have any questions about the solution? Here are five related questions you might want to explore:

1. How can you use the Pythagorean Theorem for non-right triangles?
2. What are the properties of right triangles?
3. How do you simplify radicals in different situations?
4. What is the significance of the Pythagorean triplets?
5. Can you apply the same method to isosceles right triangles?

**Tip:** Always check if a square root can be simplified by factoring out perfect squares, which helps simplify the result.

Find the length of the third side of a right triangle with legs of lengths 6 and 3√13. Write the answer in simplest radical form.

Learn how to find the hypotenuse of a right triangle with legs of lengths 6 and 3√13 using the Pythagorean Theorem. This problem is suitable for high school students in Grades 9-10 and involves simplifying radicals.

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