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To find the length of the third side of a right triangle when two sides are given, we can use the Pythagorean theorem. The Pythagorean theorem states:

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

where \( c \) is the hypotenuse (the side opposite the right angle), and \( a \) and \( b \) are the other two sides.

From the diagram:
- One leg (\( a \)) = 9 units
- The hypotenuse (\( c \)) = 15 units

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

Using the Pythagorean theorem:

\[ 9^2 + b^2 = 15^2 \]

Calculating the squares:

\[ 81 + b^2 = 225 \]

Subtracting 81 from both sides:

\[ b^2 = 144 \]

Taking the square root of both sides:

\[ b = \sqrt{144} \]
\[ b = 12 \]

Thus, the length of the third side is 12 units.

Would you like more details on this calculation, or do you have any other questions?

Here are some related questions to explore:

1. How would the Pythagorean theorem change if the triangle was not a right triangle?
2. What is the converse of the Pythagorean theorem?
3. How do you apply the Pythagorean theorem in three-dimensional space?
4. How can trigonometric ratios be used to find the sides of a right triangle?
5. What is the relationship between the Pythagorean theorem and Euclidean geometry?
6. How can the Pythagorean theorem be applied in real-life scenarios?
7. Can the Pythagorean theorem be used to prove that a triangle is a right triangle?
8. What are some common mistakes to avoid when using the Pythagorean theorem?

**Tip:** Always double-check the given sides of a triangle to ensure correct identification of the hypotenuse when using the Pythagorean theorem.

Learn how to find the length of the third side of a right triangle when two sides are given using the Pythagorean theorem. This problem involves calculating the square root and applying the Pythagorean theorem, suitable for students in Grades 7-9.

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