To find the length of the third side of the right triangle given in the image, we will use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse $c$ is equal to the sum of the squares of the other two sides $a$ and $b$. This is written as:

$c^2 = a^2 + b^2$

In the given triangle:

• One leg $a$ is $6$.
• The hypotenuse $c$ is $3\sqrt{3}$.

We need to find the length of the other leg $b$.

1. Write down the Pythagorean theorem with the given values: $(3\sqrt{3})^2 = 6^2 + b^2$

2. Simplify the equation: $9 \times 3 = 36 + b^2$ $27 = 36 + b^2$

3. Solve for $b^2$: $b^2 = 27 - 36$ $b^2 = -9$

However, this result is incorrect because the length of a side cannot be a negative number. There must be an error in the given dimensions or in the interpretation of the triangle. Let me double-check the provided values.

Based on the configuration, it appears there might be a misunderstanding or incorrect values. Let's correct the calculation as the image shows a right triangle with $a = 6$ and $c = 3\sqrt{3}$, which would indeed yield a correct value.

Upon reviewing the typical results, let's check a different configuration:

1. Recheck with the same given: $(3\sqrt{3})^2 = 6^2 + b^2$

2. Verify each term again: $27 = 36 + b^2$

Upon confirming the correct values should verify,

Would you provide more context or additional values in the image?

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Here are eight questions related to this problem:

1. What is the Pythagorean theorem and how is it applied to right triangles?
2. How do you simplify expressions involving square roots?
3. What are the steps to isolate $b$ in the Pythagorean theorem?
4. How can you check if the given dimensions of a triangle are valid?
5. What are the possible errors that can occur when interpreting the sides of a triangle?
6. Can you have a negative value under the square root when calculating the length of a side in a triangle?
7. What are the properties of right triangles involving special angle measures?
8. How does the relationship between the sides change if the given dimensions are altered?

Tip: Always double-check the values and configuration of a triangle before applying the Pythagorean theorem to avoid mistakes.