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The problem shows a right triangle where one leg has a length of \( 3\sqrt{3} \) and the hypotenuse has a length of 6. We are tasked with finding the length of the third side, which is the other leg of the triangle.

### Using the Pythagorean theorem:
The Pythagorean theorem states that for a right triangle:

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

Where:
- \( a \) and \( b \) are the lengths of the legs, and
- \( c \) is the length of the hypotenuse.

Given:
- One leg \( a = 3\sqrt{3} \),
- Hypotenuse \( c = 6 \).

We need to find the length of the other leg, \( b \).

### Step 1: Apply the Pythagorean theorem

\[
(3\sqrt{3})^2 + b^2 = 6^2
\]

Simplify both sides:

\[
(3\sqrt{3})^2 = 9 \times 3 = 27
\]
\[
6^2 = 36
\]

So the equation becomes:

\[
27 + b^2 = 36
\]

### Step 2: Solve for \( b^2 \)

\[
b^2 = 36 - 27
\]
\[
b^2 = 9
\]

### Step 3: Solve for \( b \)

\[
b = \sqrt{9} = 3
\]

Thus, the length of the third side is **3**.

Would you like further details or have any questions?

Here are some related questions to explore:
1. How is the Pythagorean theorem derived?
2. What are the properties of right triangles?
3. How do you rationalize radicals?
4. Can the Pythagorean theorem be applied in 3D space?
5. How do you find the area of a right triangle?

**Tip:** In any triangle problem, always identify whether it's a right triangle before applying the Pythagorean theorem.

Find the length of the third side. If necessary, write in simplest radical form. One leg is 3√3, and the hypotenuse is 6.

Solve for the missing side of a right triangle where one leg is 3√3 and the hypotenuse is 6 using the Pythagorean theorem. Learn how to simplify radicals and apply the theorem in this detailed step-by-step solution.

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